{VERSION 6 0 "IBM INTEL NT" "6.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 
0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 
0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 
0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 
102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times
" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 
261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 
0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 263 "" 1 18 0 0 0 0 0 
0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 
0 }{CSTYLE "" 260 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 
258 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 267 "" 0 1 
0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 268 "" 0 1 0 0 0 0 0 0 1 
0 0 0 0 0 0 0 }{CSTYLE "" 261 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" 258 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 
271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" 258 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" 261 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 283 "" 1 18 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" 258 290 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 
291 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 292 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 293 "" 0 1 0 0 0 0 0 0 1 0 0 
0 0 0 0 0 }{CSTYLE "" 258 294 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" 261 295 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 
296 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 297 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 300 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
301 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 302 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 303 "" 0 1 0 0 0 0 0 0 1 0 0 
0 0 0 0 0 }{CSTYLE "" 258 304 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" 261 305 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 
306 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 307 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 308 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 309 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 310 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
311 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 312 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim
es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 
1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" 
-1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 
1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 
-1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 
2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 
0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple
 Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 
1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "
" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 
2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 
0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal
" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }
1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "Integration and area I" }{TEXT 
312 0 "" }{TEXT -1 33 "II .. The area between two graphs" }}{PARA 0 "
" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "
" 0 "" {TEXT -1 18 "Version: 22.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 72 "Calculating the area of a region between two graphs .. a specia
l case   " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "
" 0 "" {TEXT -1 24 "Suppose that the graphs " }{XPPEDIT 18 0 "y=f(x)" 
"6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=g(x)" "6
#/%\"yG-%\"gG6#%\"xG" }{TEXT -1 18 " of two functions " }{XPPEDIT 18 
0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x)" "
6#-%\"gG6#%\"xG" }{TEXT -1 15 " are above the " }{TEXT 262 1 "x" }
{TEXT -1 11 " axis from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT 
-1 4 " to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 7 " where \+
" }{XPPEDIT 18 0 "a<b" "6#2%\"aG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 18 "Suppose also that " }{XPPEDIT 18 0 "f(x)<=g(x)" "6#1
-%\"fG6#%\"xG-%\"gG6#F'" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "a<=x" "6#
1%\"aG%\"xG" }{XPPEDIT 18 0 "``<=b" "6#1%!G%\"bG" }{TEXT -1 19 " so th
at the graph " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT 
-1 52 " is above (or at least does not go below) the graph " }
{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT -1 19 " over the
 interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 2 ". " }
}{PARA 0 "" 0 "" {TEXT -1 85 "Consider the problem of calculating the \+
area of the region bounded by the two graphs " }{XPPEDIT 18 0 "y=f(x)
" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y=g(x)" "6#
/%\"yG-%\"gG6#%\"xG" }{TEXT -1 24 " and the vertical lines " }
{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 
0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 71 ", that is, the area of the regio
n enclosed between the two graphs from " }{XPPEDIT 18 0 "x=a" "6#/%\"x
G%\"aG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }
{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 388 299 
299 {PLOTDATA 2 "62-%'CURVESG6%7[o7$$\"3++++++++]!#=$\"3]9@CMUk'y&!#<7
$$\"3!RL$e*)e&yL&F*$\"3!3;EH#eI%y&F-7$$\"3omm;z<rvcF*$\"3GQqg2X*Qy&F-7
$$\"3!**\\(=#Qy'pfF*$\"3?\"*H'f1#4&y&F-7$$\"37L$3_)\\kjiF*$\"3+c7#[[7x
y&F-7$$\"3w+DJv6C%f'F*$\"3!*)z_o_9Bz&F-7$$\"3GnmTlt$[#pF*$\"3)*Q%4'*y-
')z&F-7$$\"3-nm\"zb7/f(F*$\"3p'4$fxA,;eF-7$$\"3%HL3xQCGD)F*$\"3%o5(HK0
()QeF-7$$\"3%omT&[\\'p'))F*$\"3W2!H^=:S'eF-7$$\"3g+](o#>(G]*F*$\"3ah1M
V$=H*eF-7$$\"3gmTN$y_g,\"F-$\"3<Do,[H`CfF-7$$\"3))*\\PHY2;3\"F-$\"39\\
i[JsJcfF-7$$\"3YLLe3&Q!\\6F-$\"3cvmlu7q()fF-7$$\"3pm;H5=V37F-$\"3m1SPE
\\(H,'F-7$$\"37++v+YHv7F-$\"3=mU[>IUPgF-7$$\"37++DO>VU8F-$\"3A-dfPNTcg
F-7$$\"3;++vn:yu8F-$\"3CHnm9)GK1'F-7$$\"3>++D*>JrS\"F-$\"3A0_SaPPogF-7
$$\"3c$3-Qw2lV\"F-$\"3#e:Y87?:2'F-7$$\"3smTNGV)eY\"F-$\"3+^(eII`J2'F-7
$$\"3&*\\(o\\!f\"3]\"F-$\"3%*QACs21tgF-7$$\"3TLLe\"[Zd`\"F-$\"3o=B8#3.
22'F-7$$\"3GL$e*3\"R`c\"F-$\"3m)=MUq1p1'F-7$$\"3OLLLO2$\\f\"F-$\"3+$4i
y-^91'F-7$$\"33+v=J\\xj;F-$\"3K%RAwM&QUgF-7$$\"3ILL$)3PrC<F-$\"3(*=*p!
\\))H=gF-7$$\"3E+v=1Kd\"z\"F-$\"3wlVZD1j%)fF-7$$\"3;+Dce#R_&=F-$\"3s9(
\\7@&HYfF-7$$\"3`L$3_5o;#>F-$\"3PV4b@P#3!fF-7$$\"3[L3FB0n#)>F-$\"3A[W[
'yK_&eF-7$$\"3ym;Hs)p%[?F-$\"34SQT,CE.eF-7$$\"3hmTN.p\"o6#F-$\"3s$3qE+
wxu&F-7$$\"3%)*\\7[>8j<#F-$\"3M2lFt9b*p&F-7$$\"3$pmT&>3dSAF-$\"3+u_e%R
1*[cF-7$$\"34++]L_&pI#F-$\"33D5)[;]&*f&F-7$$\"39+]PJ%**=P#F-$\"3i[M>?/
ZbbF-7$$\"3/+v=#GOZV#F-$\"3B6HwXa#y^&F-7$$\"3/+]i\"*e]/DF-$\"3Eo;A*zzF
[&F-7$$\"3[LL$)))p>nDF-$\"3I2z-SN)zX&F-7$$\"38++D;J8MEF-$\"3*4FP]e\"*)
QaF-7$$\"3K$3-j:gWm#F-$\"3n\"**yM'fyKaF-7$$\"3_mTN'>(y%p#F-$\"3J2YF)>a
#GaF-7$$\"3V$3_:mUzs#F-$\"3X=>K1'o]U&F-7$$\"3K++vE\")4hFF-$\"3AYh0'[yO
U&F-7$$\"3#pTNc$[H#z#F-$\"3(*)*\\-$)f%RU&F-7$$\"3^L3_W:\\BGF-$\"3m2B;B
bmDaF-7$$\"3#p;HU4,h&GF-$\"3$ez%)[3?*GaF-7$$\"3K+v$Rk5())GF-$\"3`w[WG6
bLaF-7$$\"3CLLeMUZ_HF-$\"3%[7#*yv$3YaF-7$$\"3@+vo/)G#>IF-$\"35=NK&H<JY
&F-7$$\"37nmm*Q@N3$F-$\"3g)3B]B-?[&F-7$$\"33n;zV)p#\\JF-$\"38!\\'>e!)Q
-bF-7$$\"3vL3_nQZ9KF-$\"3WM/E(o'>AbF-7$$\"3[++]$f*QuKF-$\"3-'>&z)pu(Qb
F-7$$\"3*QLepxfIM$F-$\"3kq&et-7Wb&F-7$$\"3ymmm2#zWS$F-$\"3%*od2Z;<kbF-
7$$\"3D$3F%p@APMF-$\"3G.\"3*eLRnbF-7$$\"3;+v=J^'*pMF-$\"3WOQtd82pbF-7$
$\"3K++]BVI,NF-$\"3y)f,z%)G\"pbF-7$$\"3Z+D\"e^VE`$F-$\"39w3UhhenbF-7$$
\"31]i!zv@jc$F-$\"3q)>+yYkSc&F-7$$\"33+++++++OF-$\"39s**[+]aebF--%'COL
OURG6&%$RGBG$\"\"\"\"\"!$Fd_lFd_lFe_l-%*THICKNESSG6#\"\"#-F$6%7S7$F($
\"3Q0dloLdo@F-7$F4$\"3A(e'Q-0<UAF-7$F>$\"3![@UUQmoI#F-7$FH$\"3_7n'4f'H
zBF-7$FM$\"35U+mt.t]CF-7$FR$\"3T8L'o$z@>DF-7$FW$\"3!Hc%>;ZXzDF-7$Ffn$
\"3fS.QE-mPEF-7$F[o$\"3qIGYK8e#p#F-7$F`o$\"3UL')fv2FTFF-7$Feo$\"37]8g;
3W%y#F-7$Fjo$\"3)[)\\>^PI;GF-7$F_p$\"3#\\!)[JzM]%GF-7$Fdp$\"3Q+>9[+@mG
F-7$F^q$\"3MT0;<VXzGF-7$Fhq$\"35&**HHdDc)GF-7$Fbr$\"3y()*)e_a<')GF-7$F
\\s$\"3W-:r\"Q$Q\")GF-7$Fas$\"3)zG%z'=/0(GF-7$Ffs$\"3_8$fy'>(o&GF-7$F[
t$\"3RqQ&G:O&QGF-7$F`t$\"3!y:$oL-v=GF-7$Fet$\"3!=01%4lv'z#F-7$Fjt$\"3%
))4$Gf;HwFF-7$F_u$\"3;9h&=&H%\\v#F-7$Fdu$\"3*3U=L%GkMFF-7$Fiu$\"3I'[uT
&[N>FF-7$F^v$\"3$=:M$p#3hq#F-7$Fcv$\"3-k^SMSq'p#F-7$Fhv$\"3ASK9[)3Bp#F
-7$F]w$\"3u'pLo6&*Hp#F-7$Fbw$\"3KRK5wEv*p#F-7$Fgw$\"3i@)p<e48r#F-7$F\\
x$\"3S'Gl%4!3$HFF-7$Ffx$\"33Et0$R]/v#F-7$F`y$\"3]CTF+SUyFF-7$Fjy$\"3Z?
1f`z')3GF-7$Fdz$\"3y\"f^\">dGWGF-7$Fiz$\"3q&G,ME)o\")GF-7$F^[l$\"3-YA*
pzQG#HF-7$Fc[l$\"3O(*HD;8TjHF-7$Fh[l$\"3lRg*=e!z/IF-7$F]\\l$\"3z7'Q$f_
jWIF-7$Fb\\l$\"3+G3<R#y#zIF-7$Fg\\l$\"36qaUT4f:JF-7$F\\]l$\"3e,^SCE5WJ
F-7$Ff]l$\"3wy98*)o\\pJF-7$F`^l$\"3[JAj\\!R#)=$F-7$Fj^l$\"3_k$\\j>I;?$
F--F__l6&Fa_lFe_lFe_lFb_lFf_l-F$6$7$7$$\"3a**************pF*$\"3A+++A$
fuQ#F-7$Ffil$\"3:+++*=f-!eF--%&COLORG6&Fa_l$\"\"%!\"\"F`jlF`jl-F$6$7$7
$$\"3!**************R$F-$\"3\"******fyn@9$F-7$Fgjl$\"3i*****Rn<Oc&F-F]
jl-F$6%7$7$FfilFe_lFeil-F^jl6&Fa_l$\"\"&FbjlFd[mFd[m-%*LINESTYLEG6#\"
\"$-F$6%7$7$FgjlFe_lFfjlFb[mFf[m-F$6&7$7$Fe_lFe_l7$$FajlFd_lFe_l7%7$$
\"++++?R!\"*$\"+++++]!#6Fb\\m7$Ff\\m$!+++++]F[]m-%&STYLEG6#%,PATCHNOGR
IDG-F__l6&Fa_lFd_lFd_lFd_l-%)POLYGONSG6<7&7$$\"+++++q!#5$\"+*=f-!eFh\\
m7$$\"+Dj/x\")F\\^m$\"+e+,OeFh\\m7$F`^m$\"+p(\\:^#Fh\\m7$Fj]m$\"+@$fuQ
#Fh\\m7&F_^m7$$\"+R!)=,#*F\\^m$\"+O&>*yeFh\\m7$F\\_m$\"+-di5EFh\\mFd^m
7&F[_m7$$\"+^UHN5Fh\\m$\"+9=!R$fFh\\m7$Fe_m$\"+c\"Gvq#Fh\\mF`_m7&Fd_m7
$$\"+bJB^6Fh\\m$\"+H@o))fFh\\m7$F^`m$\"+[Ds&y#Fh\\mFi_m7&F]`m7$$\"+A5i
m7Fh\\m$\"+-n`MgFh\\m7$Fg`m$\"+Q'Q<%GFh\\mFb`m7&Ff`m7$$\"+Y.gt8Fh\\m$
\"+H(3I1'Fh\\m7$F`am$\"+!>aM(GFh\\mF[am7&F_am7$$\"+R7P%[\"Fh\\m$\"+fMQ
tgFh\\m7$Fiam$\"+i^Y')GFh\\mFdam7&Fham7$$\"+b1$*)f\"Fh\\m$\"+sjeggFh\\
m7$Fbbm$\"+,'**3)GFh\\mF]bm7&Fabm7$$\"+xE78<Fh\\m$\"+4IQBgFh\\m7$F[cm$
\"+XtrfGFh\\mFfbm7&Fjbm7$$\"+3KeI=Fh\\m$\"+V#4='fFh\\m7$Fdcm$\"+c&=m#G
Fh\\mF_cm7&Fccm7$$\"+3D/M>Fh\\m$\"+S]$=*eFh\\m7$F]dm$\"+&z&f#z#Fh\\mFh
cm7&F\\dm7$$\"+vJ^]?Fh\\m$\"+$e<;!eFh\\m7$Ffdm$\"+uHIaFFh\\mFadm7&Fedm
7$$\"+$3iu;#Fh\\m$\"+FLn1dFh\\m7$F_em$\"+*yg9s#Fh\\mFjdm7&F^em7$$\"+DS
;!G#Fh\\m$\"+fy,>cFh\\m7$Fhem$\"+4J$**p#Fh\\mFcem7&Fgem7$$\"+&=3DQ#Fh
\\m$\"+iTu[bFh\\m7$Fafm$\"+U\\2#p#Fh\\mF\\fm7&F`fm7$$\"+!H0U]#Fh\\m$\"
+mV\"H[&Fh\\m7$Fjfm$\"+#y4(*p#Fh\\mFefm7&Fifm7$$\"+-()H2EFh\\m$\"+_uhX
aFh\\m7$Fcgm$\"+cHS@FFh\\mF^gm7&Fbgm7$$\"+[3AFFFh\\m$\"+8'=^U&Fh\\m7$F
\\hm$\"+O&HNw#Fh\\mFggm7&F[hm7$$\"+nAPLGFh\\m$\"+#>*\\EaFh\\m7$Fehm$\"
+Re,9GFh\\mF`hm7&Fdhm7$$\"+)>P)\\HFh\\m$\"+SY[XaFh\\m7$F^im$\"+Z!)4!)G
Fh\\mFihm7&F]im7$$\"+a$R21$Fh\\m$\"+e#=^Z&Fh\\m7$Fgim$\"+`I**[HFh\\mFb
im7&Ffim7$$\"+>TXwJFh\\m$\"+*f\"y5bFh\\m7$F`jm$\"+K*3;-$Fh\\mF[jm7&F_j
m7$$\"+&R;FG$Fh\\m$\"+i-*3a&Fh\\m7$Fijm$\"+9a*Q3$Fh\\mFdjm7&Fhjm7$$\"+
++++MFh\\m$\"+twhjbFh\\m7$Fb[n$\"+'yn@9$Fh\\mF][n-F^jl6&Fa_l$\"#&)!\"#
F[\\nF[\\nF_]m-%%TEXTG6%7$$\"#CFbjl$\"\"'Fd_lQ)y~=~f(x)6\"F^_l-F_\\n6%
7$Fb\\n$\"$X#F]\\nQ)y~=~g(x)Fg\\nF`il-F_\\n6%7$$\"\"(Fbjl$F]\\nFbjlQ$x
=aFg\\nFc]m-F_\\n6%7$$\"#MFbjlFc]nQ$x=bFg\\nFc]m-F_\\n6%7$$\"$.%F]\\n$
!#9F]\\nQ\"xFg\\nFc]m-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Fb^nQ\"yFg
\\n-%%FONTG6#%(DEFAULTG-%%VIEWG6$;Fe_lF^^n;Fc]n$\"#hFbjl" 1 2 0 1 10 
0 2 9 1 1 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curv
e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curv
e 10" "Curve 11" "Curve 12" "Curve 13" }}{TEXT -1 1 " " }}{PARA 257 "
" 0 "" {TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 
278 1 "y" }{TEXT -1 49 " axis can lie anywhere in relation to the grap
hs " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 5 " and
 " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT -1 37 ", and
 so is omitted from the picture." }}{PARA 0 "" 0 "" {TEXT -1 45 "The a
rea of the region under the upper graph " }{XPPEDIT 18 0 "y=f(x)" "6#/
%\"yG-%\"fG6#%\"xG" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "x=a" "6#/%\"x
G%\"aG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }
{TEXT -1 13 " is given by " }{XPPEDIT 18 0 "Int(f(x),x=a..b)" "6#-%$In
tG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 53 ", while the area of the
 region under the lower graph " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"g
G6#%\"xG" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }
{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 13 " \+
is given by " }{XPPEDIT 18 0 "Int(g(x),x=a..b)" "6#-%$IntG6$-%\"gG6#%
\"xG/F);%\"aG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "It
 follows that " }{TEXT 259 54 "the area of the region enclosed between
 the two graphs" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%
\"aG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT 
-1 11 " is given: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"Int(f(x),x = a .. b)-Int(g(x),x = a .. b)" "6#,&-%$IntG6$-%\"fG6#%\"x
G/F*;%\"aG%\"bG\"\"\"-F%6$-%\"gG6#F*/F*;F-F.!\"\"" }{TEXT -1 14 " ----
--- (i). " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 265 13 "_________
____" }{TEXT -1 17 "                 " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "F(x)" 
"6#-%\"FG6#%\"xG" }{TEXT -1 26 " is an anti-derivative of " }{XPPEDIT 
18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "G(x
)" "6#-%\"GG6#%\"xG" }{TEXT -1 26 " is an anti-derivative of " }
{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 17 "If the integrals " }{XPPEDIT 18 0 "Int(f(x),x = a ..
 b)" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "Int(g(x),x = a .. b)" "6#-%$IntG6$-%\"gG6#%\"xG/F);%\"a
G%\"bG" }{TEXT -1 89 " are evaluated by means of the Fundamental Theor
em of Calculus by using anti-derivatives " }{XPPEDIT 18 0 "F(x)" "6#-%
\"FG6#%\"xG" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG
" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "G(x)" "6#-%\"GG6#%\"xG" }{TEXT 
-1 4 " of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 10 ", we
 have " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x \+
= a .. b)-Int(g(x),x = a .. b)" "6#,&-%$IntG6$-%\"fG6#%\"xG/F*;%\"aG%
\"bG\"\"\"-F%6$-%\"gG6#F*/F*;F-F.!\"\"" }{TEXT -1 1 " " }}{PARA 257 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=F(x)" "6#/%!G-%\"FG6#%\"xG" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([b,``],[a,``])" "6#-%*PIECEWI
SEG6$7$%\"bG%!G7$%\"aGF(" }{XPPEDIT 18 0 " ``- G(x)" "6#,&%!G\"\"\"-%
\"GG6#%\"xG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([b,``],[a,
``])" "6#-%*PIECEWISEG6$7$%\"bG%!G7$%\"aGF(" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "``=F(b)-F(a) - (G(b)-G(a))" "6#/%!G,(-%\"FG6#%\"bG\"\"\"-F'6#%\"
aG!\"\",&-%\"GG6#F)F*-F16#F-F.F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 53 "This last expression can be re-arranged in the form: " }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(b)-G(b) - (F(a)-G(a
))" "6#,(-%\"FG6#%\"bG\"\"\"-%\"GG6#F'!\"\",&-F%6#%\"aGF(-F*6#F0F,F," 
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 76 "This is the result which is obtained if we evaluated the \+
definite integral  " }{XPPEDIT 18 0 "Int(``(f(x)-g(x)),x = a .. b);" "
6#-%$IntG6$-%!G6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F-!\"\"/F-;%\"aG%\"bG" }
{TEXT -1 67 "  by the Fundamental Theorem of Calculus using the anti-d
erivative " }{XPPEDIT 18 0 "F(x)-G(x)" "6#,&-%\"FG6#%\"xG\"\"\"-%\"GG6
#F'!\"\"" }{TEXT -1 5 " of  " }{XPPEDIT 18 0 "f(x)-g(x)" "6#,&-%\"fG6#
%\"xG\"\"\"-%\"gG6#F'!\"\"" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(f(x)-g(x)),x = a .. b) = F(x)-G(
x);" "6#/-%$IntG6$-%!G6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F.!\"\"/F.;%\"aG%
\"bG,&-%\"FG6#F.F/-%\"GG6#F.F3" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEW
ISE([b, ``],[a, ``])" "6#-%*PIECEWISEG6$7$%\"bG%!G7$%\"aGF(" }{TEXT 
-1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=F(b)-G(
b)-(F(a)-G(a))" "6#/%!G,(-%\"FG6#%\"bG\"\"\"-%\"GG6#F)!\"\",&-F'6#%\"a
GF*-F,6#F2F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 49 "This p
rovides an indirect argument to show that: " }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x = a .. b)-Int(g(x),x = a .. \+
b) = Int(``(f(x)-g(x)),x = a .. b);" "6#/,&-%$IntG6$-%\"fG6#%\"xG/F+;%
\"aG%\"bG\"\"\"-F&6$-%\"gG6#F+/F+;F.F/!\"\"-F&6$-%!G6#,&-F)6#F+F0-F46#
F+F8/F+;F.F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 75 "Hence, a
t least in the special situation under discussion, where the graph " }
{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 20 " is above
 the graph " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT 
-1 6 " from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to " 
}{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 32 ", and both graphs a
re above the " }{TEXT 264 1 "x" }{TEXT -1 7 " axis, " }{TEXT 259 54 "t
he area of the region enclosed between the two graphs" }{TEXT -1 6 " f
rom " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to " }
{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 34 " is given by the sin
gle integral: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int
(``(f(x)-g(x)),x = a .. b);" "6#-%$IntG6$-%!G6#,&-%\"fG6#%\"xG\"\"\"-%
\"gG6#F-!\"\"/F-;%\"aG%\"bG" }{TEXT -1 15 " ------- (ii). " }}{PARA 
257 "" 0 "" {TEXT -1 1 " " }{TEXT 263 11 "___________" }{TEXT -1 18 " \+
                 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 72 "The formula (ii) often requires less work to evaluate tha
n formula (i). " }}{PARA 0 "" 0 "" {TEXT -1 93 "There is a better, and
 more general way of looking at this, which we shall consider shortly.
 " }}{PARA 0 "" 0 "" {TEXT -1 117 "However, for the present, in the ne
xt section we consider some examples which fit in with the scheme just
 discussed. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 
"" {TEXT -1 64 "Examples of calculating the area of a region between t
wo graphs " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 268 8 "Question" }{TEXT 
-1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 65 "Calculate the area of the reg
ion enclosed between the graphs of  " }{XPPEDIT 18 0 "y = 11+3*x-x^2;
" "6#/%\"yG,(\"#6\"\"\"*&\"\"$F'%\"xGF'F'*$F*\"\"#!\"\"" }{TEXT -1 7 "
  and  " }{XPPEDIT 18 0 "y = x^2+1;" "6#/%\"yG,&*$%\"xG\"\"#\"\"\"F)F)
" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" }
{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 1 
"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 8 "Sol
ution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 42 "This region can
 be illustrated as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 660 "f := x -> 11+3*x-x^2:\ng := x -> x
^2+1:\na := -1: b := 2:\nc := -1.4: d := 2.6:\np1 := plot([f(x),g(x)],
x=c..d,color=[red,blue],thickness=2):\np2 := plot([[[a,g(a)],[a,f(a)]]
,[[b,g(b)],[b,f(b)]]],\n                color=COLOR(RGB,.4,.4,.4)):\np
3 := plot([[[a,0],[a,g(a)]],[[b,0],[b,g(b)]]],\n                color=
COLOR(RGB,.5,.5,.5),linestyle=3):\npp := plot(f(x),x=a..b,adaptive=fal
se,numpoints=25):\nfpts := op(1,op(1,pp)):\npp := plot(g(x),x=a..b,ada
ptive=false,numpoints=25):\ngpts := op(1,op(1,pp)):\np4 := plots[polyg
onplot]([seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2..25)],\n       \+
    style=patchnogrid,color=COLOR(RGB,.9,.9,.9)):\nplots[display]([p1,
p2,p3,p4]);" }}{PARA 13 "" 1 "" {GLPLOT2D 282 369 369 {PLOTDATA 2 "6+-
%'CURVESG6%7S7$$!3!**************R\"!#<$\"3u++++++S[F*7$$!3DLLL$Q6GJ\"
F*$\"3.?/z@@4Q`F*7$$!3Ymm;M!\\pB\"F*$\"3Y_TP%)*4\"fdF*7$$!3CLLL))Qj^6F
*$\"3'zCUAsP)=iF*7$$!39LLL=Kvl5F*$\"3;V@hA/\"pm'F*7$$!3tmmmTs!G!)*!#=$
\"3svsKHv?)4(F*7$$!3tKLL3yO5!*FD$\"37.b3xB-&[(F*7$$!3u*****\\nU)*=)FD$
\"39)[(3n>JsyF*7$$!35LLL3WDTtFD$\"3]i7i9Noe#)F*7$$!33+++vvQ&\\'FD$\"3F
c`+IJ[H')F*7$$!3Elmmm&4`i&FD$\"3#\\o\"z_g'f**)F*7$$!3MKLL$QW*e[FD$\"3O
MnzzKA1$*F*7$$!3))*******p)>'*RFD$\"3X$=,&\\VWT'*F*7$$!3P*******\\5*HJ
FD$\"3!*)*)>EXjI'**F*7$$!3A*******H\"3&H#FD$\"3K!REG;!)e-\"!#;7$$!3gKL
L3k(p`\"FD$\"3mE-F6%G:0\"F^p7$$!3KjmmmO;bj!#>$\"3!*)pZziI03\"F^p7$$\"3
;HLLL`Q\"G\"Fgp$\"31vi^Ox#Q5\"F^p7$$\"3*)*****\\s]k,\"FD$\"35ZOn\\.YH6
F^p7$$\"3LKLLLvv-=FD$\"3+5gF\"zK3:\"F^p7$$\"3u,++D2YlEFD$\"3a#[t39fG<
\"F^p7$$\"3;,++v\"ep[$FD$\"3i+e$z')\\C>\"F^p7$$\"3rMLL$e/TM%FD$\"3Mi!p
G*=X67F^p7$$\"3-MLLeDBJ^FD$\"3\"**H#=HugF7F^p7$$\"3=nmm;kD!)fFD$\"3E<4
pDUkV7F^p7$$\"3Gkmm\"f`@'oFD$\"3GwV$eXv(e7F^p7$$\"3$))****\\nZ)HwFD$\"
3)pH1Z&3oq7F^p7$$\"3Wlmm;$y*e%)FD$\"3/j\"Q`.:AG\"F^p7$$\"3[*******R^bJ
*FD$\"3#z&R6Wqo#H\"F^p7$$\"3n*****\\5a`,\"F*$\"3(['fkN=^,8F^p7$$\"3w**
**\\7RV'4\"F*$\"3D%*>ITMr38F^p7$$\"3s*****\\@fk=\"F*$\"3PeTh<#p^J\"F^p
7$$\"39LLL`4Nn7F*$\"3?4&3@W(e>8F^p7$$\"3++++],s`8F*$\"3p(R[0AgGK\"F^p7
$$\"3dmm;zM)>V\"F*$\"3'GP*Gvt`C8F^p7$$\"3-+++qfa<:F*$\"3'en$*Q@p\\K\"F
^p7$$\"3+LL$eg`!)f\"F*$\"3M3$Q!\\&QSK\"F^p7$$\"3%)****\\#G2Ao\"F*$\"3^
'R?10!o@8F^p7$$\"3ELLL$)G[k<F*$\"3zoBW!)[+=8F^p7$$\"3!*****\\7yh]=F*$
\"3erd&\\r1FJ\"F^p7$$\"3'ommm)fdL>F*$\"3f-'yj=,iI\"F^p7$$\"3kmmm,FT=?F
*$\"3!p1vq#[7)H\"F^p7$$\"3OLL$e#pa-@F*$\"3qzo@?Pp)G\"F^p7$$\"3)*******
Rv&)z@F*$\"3+:0`s$z(y7F^p7$$\"3SLLLGUYoAF*$\"3e0s(HFYfE\"F^p7$$\"3gmmm
1^rZBF*$\"3?7Hz4z8`7F^p7$$\"3=++]sI@KCF*$\"3.6guyy4Q7F^p7$$\"3=++]2%)3
8DF*$\"3H6*ey=lBA\"F^p7$$\"33+++++++EF*$\"3#************R?\"F^p-%'COLO
URG6&%$RGBG$\"\"\"\"\"!$F`[lF`[lFa[l-%*THICKNESSG6#\"\"#-F$6%7S7$F($\"
3'*************fHF*7$F.$\"3+!e4#GPZBFF*7$F3$\"31\\e78H/IDF*7$F8$\"3`_x
v71EEBF*7$F=$\"3'p&yQA*He8#F*7$FB$\"3[CF<)H]4'>F*7$FH$\"3T'\\9/Gn=\"=F
*7$FM$\"3O6DTI_tq;F*7$FR$\"3.O(yG;S*Q:F*7$FW$\"3#Rk%\\(f+>U\"F*7$Ffn$
\"30:$3s2TkJ\"F*7$F[o$\"3]lK?0M4O7F*7$F`o$\"3!o\"))\\Sgpf6F*7$Feo$\"3S
-,Q(Rjz4\"F*7$Fjo$\"3%o4O<)Rn_5F*7$F`p$\"3TKxzkHiB5F*7$Fep$\"3.7I_5)QS
+\"F*7$F[q$\"3<[s$[>k,+\"F*7$F`q$\"3GINw?<L55F*7$Feq$\"3*3!*RsM*\\K5F*
7$Fjq$\"3Wv^w3o/r5F*7$F_r$\"3N$*>9t()e@6F*7$Fdr$\"3yw$4jW7()=\"F*7$Fir
$\"3_+qnvaHj7F*7$F^s$\"3WG34oYjd8F*7$Fcs$\"31Pi:>:*3Z\"F*7$Fhs$\"3%)Gq
Vbd9#e\"F*7$F]t$\"3=o$=;9Vbr\"F*7$Fbt$\"3M>/')y\\zn=F*7$Fgt$\"3P].aeR%
4.#F*7$F\\u$\"3Dd+[Ct;-AF*7$Fau$\"3]:%e)oao2CF*7$Ffu$\"3I3\\\"*Q%yhg#F
*7$F[v$\"3BAg^W#eD$GF*7$F`v$\"3=rig%ow00$F*7$Fev$\"30TK1rd%HI$F*7$Fjv$
\"3Y<p6F`x`NF*7$F_w$\"3ZMgHT8#)HQF*7$Fdw$\"3A8jdX)*R8TF*7$Fiw$\"3N%GUz
G'yCWF*7$F^x$\"3$Q(R@'4;(QZF*7$Fcx$\"3#>L\\U$)*)R2&F*7$Fhx$\"3&4?J`d.2
U&F*7$F]y$\"3Q^[p%*)y<v&F*7$Fby$\"3QXzAb*Hf9'F*7$Fgy$\"3&y(32Aiw6lF*7$
F\\z$\"3t!*)R+Vgc\"pF*7$Faz$\"3D()3\"RM8cJ(F*7$Ffz$\"3o++++++gxF*-F[[l
6&F][lFa[lFa[lF^[lFb[l-F$6$7$7$$!\"\"F`[l$Fe[lF`[l7$Fbel$\"\"(F`[l-%&C
OLORG6&F][l$\"\"%FcelF[flF[fl-F$6$7$7$Fdel$\"\"&F`[l7$Fdel$\"#8F`[lFhe
l-F$6%7$7$FbelFa[lFael-Fiel6&F][l$FbflFcelF\\glF\\gl-%*LINESTYLEG6#\"
\"$-F$6%7$7$FdelFa[lF`flFjflF]gl-%)POLYGONSG6<7&Feel7$$!+]2<#p)!#5$\"+
_/\"oj(!\"*7$Fjgl$\"+B$Qbv\"F_hlFael7&Figl7$$!+8bBavF\\hl$\"+/Y1j\")F_
hl7$Fehl$\"+UZmq:F_hlF`hl7&Fdhl7$$!+D$3XF'F\\hl$\"+bH&Rs)F_hl7$F^il$\"
+ZXp$R\"F_hlFihl7&F]il7$$!+v#)H')\\F\\hl$\"+8)yaD*F_hl7$Fgil$\"+0<j[7F
_hlFbil7&Ffil7$$!+j3@/PF\\hl$\"+g\\_^(*F_hl7$F`jl$\"+\"y6s8\"F_hlF[jl7
&F_jl7$$!+8<b:DF\\hl$\"+[a?=5!\")7$Fijl$\"+/+Gj5F_hlFdjl7&Fhjl7$$!+8Sw
%G\"F\\hl$\"+hk!)f5F][m7$Fc[m$\"+'=1l,\"F_hlF^[m7&Fb[m7$$!+]7;)=\"!#7$
\"+5Mk*4\"F][m7$F\\\\m$\"+79++5F_hlFg[m7&F[\\m7$$\"+P'=pD\"F\\hl$\"+:x
7O6F][m7$Ff\\m$\"+X%)z:5F_hlFa\\m7&Fe\\m7$$\"+]c.iDF\\hl$\"+VqHq6F][m7
$F_]m$\"+n-kl5F_hlFj\\m7&F^]m7$$\"+DMe6PF\\hl$\"+^;d(>\"F][m7$Fh]m$\"+
:&ex8\"F_hlFc]m7&Fg]m7$$\"+]>q0]F\\hl$\"+1S6D7F][m7$Fa^m$\"+?0d]7F_hlF
\\^m7&F`^m7$$\"+]U80jF\\hl$\"+5$*R\\7F][m7$Fj^m$\"+zra(R\"F_hlFe^m7&Fi
^m7$$\"+]!ytb(F\\hl$\"+ztgp7F][m7$Fc_m$\"+I'R6d\"F_hlF^_m7&Fb_m7$$\"+(
QNXp)F\\hl$\"+;6C&G\"F][m7$F\\`m$\"+c%\\fv\"F_hlFg_m7&F[`m7$$\"+XDn/5F
_hl$\"+r]Y+8F][m7$Fe`m$\"+BpO4?F_hlF``m7&Fd`m7$$\"+!y?#>6F_hl$\"+>2]58
F][m7$F^am$\"+a^l_AF_hlFi`m7&F]am7$$\"+4wY_7F_hl$\"+sF()=8F][m7$Fgam$
\"+6^noDF_hlFbam7&Ffam7$$\"+IOTq8F_hl$\"+P2KB8F][m7$F`bm$\"+<N.yGF_hlF
[bm7&F_bm7$$\"+4\">)*\\\"F_hl$\"+(****\\K\"F][m7$Fibm$\"+ftX\\KF_hlFdb
m7&Fhbm7$$\"+EP/B;F_hl$\"+Cg[B8F][m7$Fbcm$\"+P4FMOF_hlF]cm7&Facm7$$\"+
)o:;v\"F_hl$\"+b*o'=8F][m7$F[dm$\"+<v:oSF_hlFfcm7&Fjcm7$$\"+%)[op=F_hl
$\"+4LL68F][m7$Fddm$\"+l:s&\\%F_hlF_dm7&FcdmFcflF`flFhdm-Fiel6&F][l$\"
\"*FcelF^emF^em-%&STYLEG6#%,PATCHNOGRIDG-%+AXESLABELSG6%Q\"x6\"Q!Fhem-
%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!#9Fcel$\"#EFcelF]fm" 1 2 0 1 10 0 2 9 
1 4 2 1.000000 45.000000 43.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "
Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 14 "The graph of  " }{XPPEDIT 18 0 "y=x^2+1" "6#/%\"yG,&*$%\"
xG\"\"#\"\"\"F)F)" }{TEXT -1 13 " is drawn in " }{TEXT 256 4 "blue" }
{TEXT -1 22 ", while the graph of  " }{XPPEDIT 18 0 "y=11+3*x-x^2" "6#
/%\"yG,(\"#6\"\"\"*&\"\"$F'%\"xGF'F'*$F*\"\"#!\"\"" }{TEXT -1 13 " is \+
drawn in " }{TEXT 260 3 "red" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "We can calculate the area
 of this region by subtracting " }{XPPEDIT 18 0 "Int(``(x^2+1),x = -1 \+
.. 2);" "6#-%$IntG6$-%!G6#,&*$%\"xG\"\"#\"\"\"F-F-/F+;,$F-!\"\"F," }
{TEXT -1 7 "  from " }{XPPEDIT 18 0 "Int(``(11+3*x-x^2),x = -1 .. 2);
" "6#-%$IntG6$-%!G6#,(\"#6\"\"\"*&\"\"$F+%\"xGF+F+*$F.\"\"#!\"\"/F.;,$
F+F1F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 
257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(x^2+1),x = -1 .. 2) \+
= x^3/3+x;" "6#/-%$IntG6$-%!G6#,&*$%\"xG\"\"#\"\"\"F.F./F,;,$F.!\"\"F-
,&*&F,\"\"$F5F2F.F,F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2,``
],[-1,``])" "6#-%*PIECEWISEG6$7$\"\"#%!G7$,$\"\"\"!\"\"F(" }{TEXT -1 
1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(8/3+2)-
(-1/3-1)" "6#/%!G,&-F$6#,&*&\"\")\"\"\"\"\"$!\"\"F+\"\"#F+F+,&*&F+F+F,
F-F-F+F-F-" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``= 6" "6#/%!G\"\"'" }{TEXT -1 2 ", " }}{PARA 258 "" 0 
"" {TEXT -1 6 "while " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "Int(``(11+3*x-x^2),x = -1 .. 2) = 11*x+3*x^2/2-x^3/3;" "6#/-%$In
tG6$-%!G6#,(\"#6\"\"\"*&\"\"$F,%\"xGF,F,*$F/\"\"#!\"\"/F/;,$F,F2F1,(*&
F+F,F/F,F,*(F.F,*$F/F1F,F1F2F,*&F/F.F.F2F2" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "PIECEWISE([2,``],[-1,``])" "6#-%*PIECEWISEG6$7$\"\"#%!G7$,$\"\"
\"!\"\"F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = ``(22+6-8/3)-(-11+3/2+1/3);" "6#/%!G,&-F$6#,(\"#A
\"\"\"\"\"'F**&\"\")F*\"\"$!\"\"F/F*,(\"#6F/*&F.F*\"\"#F/F**&F*F*F.F/F
*F/" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "``=39-9/3-3/2" "6#/%!G,(\"#R\"\"\"*&\"\"*F'\"\"$!\"\"F+*&F*F'\"\"#F
+F+" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "``=34" "6#/%!G\"#M" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/2=69/2" "6#/*
&\"\"\"F%\"\"#!\"\"*&\"#pF%F&F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" 
{TEXT -1 29 "so the required area is:  34 " }{XPPEDIT 18 0 "1/2-6 = 28
" "6#/,&*&\"\"\"F&\"\"#!\"\"F&\"\"'F(\"#G" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "1/2 = 57/2" "6#/*&\"\"\"F%\"\"#!\"\"*&\"#dF%F&F'" }{TEXT -1 2 ".
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 83 "Int(x^2+1,x=-1..2);\nA1 := value(%);\nInt(11+3*x-x^2,
x=-1..2);\nA2 := value(%);\nA2-A1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%$IntG6$,&*$)%\"xG\"\"#\"\"\"F+F+F+/F);!\"\"F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A1G\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6
$,(\"#6\"\"\"*&\"\"$F(%\"xGF(F(*$)F+\"\"#F(!\"\"/F+;F/F." }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#A2G#\"#p\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6##\"#d\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 23 "Alternatively, letting " }{XPPEDIT 18 0 "
f(x)=11+3*x-x^2" "6#/-%\"fG6#%\"xG,(\"#6\"\"\"*&\"\"$F*F'F*F**$F'\"\"#
!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x)=x^2+1" "6#/-%\"gG6#%\"
xG,&*$F'\"\"#\"\"\"F+F+" }{TEXT -1 37 ", we can obtain the required ar
ea as " }{XPPEDIT 18 0 "Int(``(f(x)-g(x)),x = -1 .. 2);" "6#-%$IntG6$-
%!G6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F-!\"\"/F-;,$F.F2\"\"#" }{TEXT -1 2 
". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Now  " }{XPPEDIT 18 0 "f(x)-g(x)=10
+3*x-2*x^2" "6#/,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F(!\"\",(\"#5F)*&\"\"$F)F
(F)F)*&\"\"#F)*$F(F3F)F-" }{TEXT -1 27 ", so the required area is: " }
}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(10+3*x-2*x^2)
,x = -1 .. 2) = 10*x+3*x^2/2-2*x^3/3;" "6#/-%$IntG6$-%!G6#,(\"#5\"\"\"
*&\"\"$F,%\"xGF,F,*&\"\"#F,*$F/F1F,!\"\"/F/;,$F,F3F1,(*&F+F,F/F,F,*(F.
F,*$F/F1F,F1F3F,*(F1F,*$F/F.F,F.F3F3" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
PIECEWISE([2, ``],[-1, ``])" "6#-%*PIECEWISEG6$7$\"\"#%!G7$,$\"\"\"!\"
\"F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "``=``(20+6-16/3)-(-10+3/2+2/3)" "6#/%!G,&-F$6#,(\"#?\"\"\"\"\"'F**&
\"#;F*\"\"$!\"\"F/F*,(\"#5F/*&F.F*\"\"#F/F**&F3F*F.F/F*F/" }{TEXT -1 
1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=36-18/3 -3
/2" "6#/%!G,(\"#O\"\"\"*&\"#=F'\"\"$!\"\"F+*&F*F'\"\"#F+F+" }{TEXT -1 
1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=28" "6#/%!
G\"#G" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/2 = 57/2" "6#/*&\"\"\"F%\"\"#
!\"\"*&\"#dF%F&F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 11 "as \+
before. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "Int(10+3*x-2*x^2,x=-1..2);\nvalue(%);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%$IntG6$,(\"#5\"\"\"*&\"\"$F(%\"xGF(F(*&\"\"#F()F
+F-F(!\"\"/F+;F/F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#d\"\"#" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 266 8 "Q
uestion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 65 "Calculate the
 area of the region enclosed between the graphs of  " }{XPPEDIT 18 0 "
y = x+3;" "6#/%\"yG,&%\"xG\"\"\"\"\"$F'" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y = x^2+1;" "6#/%\"yG,&*$%\"xG\"\"#\"\"\"F)F)" }{TEXT 
-1 6 " from " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 
4 " to " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 8 "Solution
" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 42 "This region can be i
llustrated as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 607 "f := x -> x+3:\ng := x -> x^2+1:\n
a := -1: b := 1:\nc := -1.4: d := 1.6:\np1 := plot([f(x),g(x)],x=c..d,
color=[red,blue],thickness=2):\np2 := plot([[b,g(b)],[b,f(b)]],color=C
OLOR(RGB,.4,.4,.4)):\np3 := plot([[[a,0],[a,g(a)]],[[b,0],[b,g(b)]]],
\n         color=COLOR(RGB,.5,.5,.5),linestyle=3):\npp := plot(f(x),x=
a..b,adaptive=false,numpoints=25):\nfpts := op(1,op(1,pp)):\npp := plo
t(g(x),x=a..b,adaptive=false,numpoints=25):\ngpts := op(1,op(1,pp)):\n
p4 := plots[polygonplot]([seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],i=
2..25)],\n           style=patchnogrid,color=COLOR(RGB,.9,.9,.9)):\npl
ots[display]([p1,p2,p3,p4]);" }}{PARA 13 "" 1 "" {GLPLOT2D 354 415 
415 {PLOTDATA 2 "6*-%'CURVESG6%7S7$$!3!**************R\"!#<$\"33++++++
+;F*7$$!3')****\\P&3YL\"F*$\"39++]i9Rl;F*7$$!3#)**\\iv<rx7F*$\"3=+]PC#
)GA<F*7$$!3(****\\iTDP@\"F*$\"3/++v$euiy\"F*7$$!3%)***\\P\"\\J\\6F*$\"
3;++D'3&o]=F*7$$!3))**\\7V0@&3\"F*$\"38+](oX*y9>F*7$$!3****\\i&exd-\"F
*$\"3-+]P9CAu>F*7$$!3v)**\\i+#QU'*!#=$\"3!***\\P*zhd.#F*7$$!3/***\\i!3
%f+*FN$\"33+]P>fS*4#F*7$$!3G***\\7oS:P)FN$\"3=+](=$f%G;#F*7$$!3s)****
\\<#)*=xFN$\"3-++]#y,\"GAF*7$$!3.****\\(G3U9(FN$\"3?++Dr\"zbG#F*7$$!3e
)****\\-\\r\\'FN$\"3-++](4&G]BF*7$$!3K*****\\(GVZeFN$\"3=++]7nD:CF*7$$
!3j)****\\(4J@_FN$\"3/++]-*oyZ#F*7$$!3A***\\iIKFl%FN$\"3>+]PpnsMDF*7$$
!3'*)****\\FPm(RFN$\"35++]siL-EF*7$$!3)*)******4'*QS$FN$\"35+++!R5'fEF
*7$$!3J***\\i&>mPFFN$\"32+]P/QBEFF*7$$!3)*******\\=$z9#FN$\"3++++:o?&y
#F*7$$!3Y)**\\iX/4]\"FN$\"3Q+]Pa&4*\\GF*7$$!3a$)**\\(o8y%))!#>$\"3R+]7
j=_6HF*7$$!3)=)***\\i:#>CF_r$\"3=++vVy!e(HF*7$$\"3a=+](=WU[$F_r$\"3'**
*\\(=WU[.$F*7$$\"371++DJ#>&)*F_r$\"31++DJ#>&)4$F*7$$\"3m++v$>:mk\"FN$
\"31+]P>:mkJF*7$$\"3Y++DcdQAAFN$\"3/+]iv&QAA$F*7$$\"3`,+]PPBWGFN$\"3;+
+vtLU%G$F*7$$\"3%)******\\Nm'[$FN$\"3w*****\\Nm'[LF*7$$\"3******\\(yb^
6%FN$\"3+++vyb^6MF*7$$\"3v++vVVDBZFN$\"33+]PMaKsMF*7$$\"3W++]7TW)R&FN$
\"3#)***\\7TW)RNF*7$$\"3y)*****\\@80gFN$\"3))*****\\@80g$F*7$$\"3U,++D
6!Hl'FN$\"39++]7,HlOF*7$$\"3_***\\P4w)RsFN$\"3&***\\P4w)Rs$F*7$$\"3g-+
+vZf\")yFN$\"3E++]x%f\")y$F*7$$\"3%))**\\P/-a[)FN$\"3))**\\P/-a[QF*7$$
\"3G,+v=Yb;\"*FN$\"38+](=Yb;\"RF*7$$\"3h*****\\i@Ot*FN$\"3'*****\\i@Ot
RF*7$$\"3&***\\PfL'z.\"F*$\"3R+]PfL'z.%F*7$$\"3G+++!*>=+6F*$\"3G+++!*>
=+TF*7$$\"35++DE&4Q;\"F*$\"3c++DE&4Q;%F*7$$\"3E+]P%>5pA\"F*$\"3#)**\\P
%>5pA%F*7$$\"3C+++bJ*[G\"F*$\"3o+++bJ*[G%F*7$$\"3=++Dr\"[8N\"F*$\"3i++
Dr\"[8N%F*7$$\"34+++Ijy59F*$\"3`+++Ijy5WF*7$$\"3;+]P/)fTZ\"F*$\"3g+]P/
)fTZ%F*7$$\"3:+]i0j\"[`\"F*$\"3:+]i0j\"[`%F*7$F+$\"3k*************f%F*
-%'COLOURG6&%$RGBG$\"\"\"\"\"!$F][lF][lF^[l-%*THICKNESSG6#\"\"#-F$6%7S
7$F($\"3'*************fHF*7$F.$\"3_))yO[*z6y#F*7$F3$\"3$)*yq:QZDj#F*7$
F8$\"3cN70'QHJZ#F*7$F=$\"3_p\"o4xC4K#F*7$FB$\"3!pl4H#>ox@F*7$FG$\"3?a1
<b'>A0#F*7$FL$\"3]PXa2`vH>F*7$FR$\"3'))yc!)pp5\"=F*7$FW$\"38P#yP$p#3q
\"F*7$Ffn$\"31x'z\"eo#ef\"F*7$F[o$\"3sO=b?rR5:F*7$F`o$\"3E%eIXXH@U\"F*
7$Feo$\"3g2jF7Z#>M\"F*7$Fjo$\"3Sal(H)3is7F*7$F_p$\"3RCi8z\"zk@\"F*7$Fd
p$\"3E%>p,WO\"e6F*7$Fip$\"32_ff'3le6\"F*7$F^q$\"3s&)p')Hz%\\2\"F*7$Fcq
$\"3?WCK7h8Y5F*7$Fhq$\"3%4\"z'=9FD-\"F*7$F]r$\"38r[q!QGy+\"F*7$Fcr$\"3
SCSUg_e+5F*7$Fhr$\"3#G\"ev&*R@,5F*7$F]s$\"3,\"4E*Qgq45F*7$Fbs$\"3T$)G'
fT8r-\"F*7$Fgs$\"3wGe\\%)**Q\\5F*7$F\\t$\"3KK``bm*33\"F*7$Fat$\"3'f)*3
rAo:7\"F*7$Fft$\"3]ZRbr]Mp6F*7$F[u$\"3^Kv&fJ\"4B7F*7$F`u$\"35fyN))>V\"
H\"F*7$Feu$\"3%fj*Q@hhg8F*7$Fju$\"3%GE!zL4hU9F*7$F_v$\"3[F&G&e!eT_\"F*
7$Fdv$\"3YtI(>O&>@;F*7$Fiv$\"3Wm2Wy/-?<F*7$F^w$\"3o>l;\"o:6$=F*7$Fcw$
\"3WwmQ**QVZ>F*7$Fhw$\"3SN]SNzOx?F*7$F]x$\"3zg.76/S5AF*7$Fbx$\"3#*\\-R
8EXaBF*7$Fgx$\"3\"3Ih]i3`]#F*7$F\\y$\"3)[&ew>/&4l#F*7$Fay$\"33?2%*z=9E
GF*7$Ffy$\"3=p[\"*o!=.*HF*7$F[z$\"3))Q\\$)Gr9tJF*7$F`z$\"3stB,#4hcN$F*
7$F+$\"3]++++++gNF*-Fhz6&FjzF^[lF^[lF[[lF_[l-F$6$7$7$F[[l$Fb[lF][l7$F[
[l$\"\"%F][l-%&COLORG6&Fjz$Fbel!\"\"FfelFfel-F$6%7$7$$FgelF][lF^[l7$F
\\flF_el-Fdel6&Fjz$\"\"&FgelF`flF`fl-%*LINESTYLEG6#\"\"$-F$6%7$7$F[[lF
^[lF^elF^flFbfl-%)POLYGONSG6<7&F]fl7$$!+LQ6G\"*!#5$\"+<')=(3#!\"*7$F_g
l$\"+AYAL=FdglF]fl7&F^gl7$$!+U.\\p$)Fagl$\"+m40j@Fdgl7$Fjgl$\"+'o$[+<F
dglFegl7&Figl7$$!+$))Qj^(Fagl$\"+7hO[AFdgl7$Fchl$\"+-N&\\c\"FdglF^hl7&
Fbhl7$$!+$=Kvl'Fagl$\"+#yYUL#Fdgl7$F\\il$\"+[tAV9FdglFghl7&F[il7$$!+Us
!G!eFagl$\"+w#>(>CFdgl7$Feil$\"+>dsO8FdglF`il7&Fdil7$$!+3yO5]Fagl$\"+>
K'*)\\#Fdgl7$F^jl$\"+cy.^7FdglFiil7&F]jl7$$!+vE%)*=%Fagl$\"+Ld,\"e#Fdg
l7$Fgjl$\"+;yav6FdglFbjl7&Ffjl7$$!+3WDTLFagl$\"+fX(em#Fdgl7$F`[m$\"+5)
R;6\"FdglF[[m7&F_[m7$$!+vvQ&\\#Fagl$\"+V7Y]FFdgl7$Fi[m$\"+\"fpA1\"Fdgl
Fd[m7&Fh[m7$$!+n&4`i\"Fagl$\"+V!pu$GFdgl7$Fb\\m$\"+7jTE5FdglF]\\m7&Fa
\\m7$$!+LQW*e)!#6$\"+ib59HFdgl7$F[]m$\"+byP25FdglFf\\m7&Fj\\m7$$\"+++I
,Q!#8$\"+I,Q+IFdgl7$Fe]m$\"+W,++5FdglF`]m7&Fd]m7$$\"++]*3q)F]]m$\"+]*3
q3$Fdgl7$F_^m$\"+d0d25FdglFj]m7&F^^m7$$\"++(=\\q\"Fagl$\"+q=\\qJFdgl7$
Fh^m$\"+yu1H5FdglFc^m7&Fg^m7$$\"+#fBIY#Fagl$\"+fBIYKFdgl7$Fa_m$\"+_[mg
5FdglF\\_m7&F`_m7$$\"+LO[kLFagl$\"+j$[kL$Fdgl7$Fj_m$\"+,v>86FdglFe_m7&
Fi_m7$$\"+L&Q\"GTFagl$\"+`Q\"GT$Fdgl7$Fc`m$\"+y_Tq6FdglF^`m7&Fb`m7$$\"
+D2X;]Fagl$\"+s]k,NFdgl7$F\\am$\"+zxk^7FdglFg`m7&F[am7$$\"+Lvv-eFagl$
\"+`dF!e$Fdgl7$Feam$\"+]*>nL\"FdglF`am7&Fdam7$$\"+D2YlmFagl$\"+tgamOFd
gl7$F^bm$\"+nOGW9FdglFiam7&F]bm7$$\"+v\"ep[(Fagl$\"+<ep[PFdgl7$Fgbm$\"
+Faag:FdglFbbm7&Ffbm7$$\"+$e/TM)Fagl$\"+e/TMQFdgl7$F`cm$\"+83C'p\"Fdgl
F[cm7&F_cm7$$\"+eDBJ\"*Fagl$\"+cK78RFdgl7$Ficm$\"+!3%zL=FdglFdcm7&Fhcm
F`elF^elF]dm-Fdel6&Fjz$\"\"*FgelFcdmFcdm-%&STYLEG6#%,PATCHNOGRIDG-%+AX
ESLABELSG6%Q\"x6\"Q!F]em-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!#9Fgel$\"#;Fg
elFbem" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve
 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 14 "The graph of  " }{XPPEDIT 18 0 "y=x^2+1
" "6#/%\"yG,&*$%\"xG\"\"#\"\"\"F)F)" }{TEXT -1 13 " is drawn in " }
{TEXT 256 4 "blue" }{TEXT -1 22 ", while the graph of  " }{XPPEDIT 18 
0 "y = x+3;" "6#/%\"yG,&%\"xG\"\"\"\"\"$F'" }{TEXT -1 13 " is drawn in
 " }{TEXT 260 3 "red" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 56 "We can calculate the area of this reg
ion by subtracting " }{XPPEDIT 18 0 "Int(``(x^2+1),x = -1 .. 1);" "6#-
%$IntG6$-%!G6#,&*$%\"xG\"\"#\"\"\"F-F-/F+;,$F-!\"\"F-" }{TEXT -1 7 "  \+
from " }{XPPEDIT 18 0 "Int(``(x+3),x = -1 .. 1);" "6#-%$IntG6$-%!G6#,&
%\"xG\"\"\"\"\"$F+/F*;,$F+!\"\"F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 4 "Now " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "I
nt(``(x^2+1),x = -1 .. 1) = x^3/3+x;" "6#/-%$IntG6$-%!G6#,&*$%\"xG\"\"
#\"\"\"F.F./F,;,$F.!\"\"F.,&*&F,\"\"$F5F2F.F,F." }{TEXT -1 1 " " }
{XPPEDIT 18 0 "PIECEWISE([1, ``],[-1, ``]);" "6#-%*PIECEWISEG6$7$\"\"
\"%!G7$,$F'!\"\"F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "`` = ``(1/3+1)-(-1/3-1);" "6#/%!G,&-F$6#,&*&\"\"\"F*
\"\"$!\"\"F*F*F*F*,&*&F*F*F+F,F,F*F,F," }{TEXT -1 1 " " }}{PARA 257 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2;" "6#/%!G\"\"#" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "2/3=8/3" "6#/*&\"\"#\"\"\"\"\"$!\"\"*&\"\")F&
F'F(" }{TEXT -1 2 ", " }}{PARA 258 "" 0 "" {TEXT -1 6 "while " }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(x+3),x = -1 ..
 1) = x^2/2+3*x;" "6#/-%$IntG6$-%!G6#,&%\"xG\"\"\"\"\"$F,/F+;,$F,!\"\"
F,,&*&F+\"\"#F4F1F,*&F-F,F+F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECE
WISE([1, ``],[-1, ``]);" "6#-%*PIECEWISEG6$7$\"\"\"%!G7$,$F'!\"\"F(" }
{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` =
 ``(1/2+3)-(1/2-3);" "6#/%!G,&-F$6#,&*&\"\"\"F*\"\"#!\"\"F*\"\"$F*F*,&
*&F*F*F+F,F*F-F,F," }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "`` = 6;" "6#/%!G\"\"'" }{TEXT -1 2 ", " }}{PARA 0 "
" 0 "" {TEXT -1 26 "so the required area is:  " }{XPPEDIT 18 0 "6-8/3=
10/3" "6#/,&\"\"'\"\"\"*&\"\")F&\"\"$!\"\"F**&\"#5F&F)F*" }{XPPEDIT 
18 0 "``=3" "6#/%!G\"\"$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/3" "6#*&\"
\"\"F$\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Int(x^2+1,x=-1..1);\nA1 := v
alue(%);\nInt(x+3,x=-1..1);\nA2 := value(%);\nA2-A1;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#-%$IntG6$,&*$)%\"xG\"\"#\"\"\"F+F+F+/F);!\"\"F+" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G#\"\")\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$,&%\"xG\"\"\"\"\"$F(/F';!\"\"F(" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#A2G\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##
\"#5\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Alternatively, let
ting " }{XPPEDIT 18 0 "f(x) = x+3;" "6#/-%\"fG6#%\"xG,&F'\"\"\"\"\"$F)
" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x)=x^2+1" "6#/-%\"gG6#%\"xG,&*
$F'\"\"#\"\"\"F+F+" }{TEXT -1 37 ", we can obtain the required area as
 " }{XPPEDIT 18 0 "Int(``(f(x)-g(x)),x = -1 .. 1);" "6#-%$IntG6$-%!G6#
,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F-!\"\"/F-;,$F.F2F." }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 5 "Now  " }{XPPEDIT 18 0 "f(x)-g(x) = 2+x-x^2
;" "6#/,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F(!\"\",(\"\"#F)F(F)*$F(F/F-" }
{TEXT -1 27 ", so the required area is: " }}{PARA 257 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "Int(``(2+x-x^2),x = -1 .. 1) = 2*x+x^2/2-x^3/
3;" "6#/-%$IntG6$-%!G6#,(\"\"#\"\"\"%\"xGF,*$F-F+!\"\"/F-;,$F,F/F,,(*&
F+F,F-F,F,*&F-F+F+F/F,*&F-\"\"$F7F/F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 
"PIECEWISE([1, ``],[-1, ``]);" "6#-%*PIECEWISEG6$7$\"\"\"%!G7$,$F'!\"
\"F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "`` = ``(2+1/2-1/3)-(-2+1/2+1/3);" "6#/%!G,&-F$6#,(\"\"#\"\"\"*&F*F*
F)!\"\"F**&F*F*\"\"$F,F,F*,(F)F,*&F*F*F)F,F**&F*F*F.F,F*F," }{TEXT -1 
1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 4-2/3;" 
"6#/%!G,&\"\"%\"\"\"*&\"\"#F'\"\"$!\"\"F+" }{TEXT -1 1 " " }}{PARA 
257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 3;" "6#/%!G\"\"$" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "1/3 = 10/3;" "6#/*&\"\"\"F%\"\"$!\"\"*&
\"#5F%F&F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 11 "as before.
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "Int(2+x-x^2,x=-1..1);\nvalue(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$,(\"\"#\"\"\"%\"xGF(*$)F)F'F(!\"\"/F);F,F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#5\"\"$" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 270 8 "Question" }{TEXT 
-1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 65 "Calculate the area of the reg
ion enclosed between the graphs of  " }{XPPEDIT 18 0 "y = 1/x;" "6#/%
\"yG*&\"\"\"F&%\"xG!\"\"" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "y = ex
p(x);" "6#/%\"yG-%$expG6#%\"xG" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "x
 = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 2;" "6#
/%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 271 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" 
{TEXT -1 42 "This region can be illustrated as follows." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 669 "f := x
 -> exp(x):\ng := x -> 1/x:\na := 1: b := 2:\nc := 0: d := 2.3:\np1 :=
 plot([f(x),g(x)],x=c..d,color=[red,blue],thickness=2):\np2 := plot([[
[a,g(a)],[a,f(a)]],[[b,g(b)],[b,f(b)]]],\n                color=COLOR(
RGB,.4,.4,.4)):\np3 := plot([[[a,0],[a,g(a)]],[[b,0],[b,g(b)]]],\n    \+
            color=COLOR(RGB,.5,.5,.5),linestyle=3):\npp := plot(f(x),x
=a..b,adaptive=false,numpoints=25):\nfpts := op(1,op(1,pp)):\npp := pl
ot(g(x),x=a..b,adaptive=false,numpoints=25):\ngpts := op(1,op(1,pp)):
\np4 := plots[polygonplot]([seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],
i=2..25)],\n           style=patchnogrid,color=COLOR(RGB,.9,.9,.9)):\n
plots[display]([p1,p2,p3,p4],view=[0..2.3,0..8]);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 250 420 420 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$\"\"!F)$\"\"\"F)
7$$\"3qKLLeaM8]!#>$\"3'H@%o-9T^5!#<7$$\"3WmmTN0Vv$*F/$\"3u4X#)o)*G)4\"
F27$$\"3ALL3U^5G9!#=$\"3u?@(p?6N:\"F27$$\"3VLLe%**=>#>F;$\"3oQB>eI!>@
\"F27$$\"3Hm;/OeQ8CF;$\"37`^,j>&HF\"F27$$\"3AL$3-^Q!pGF;$\"3a<kS3hHK8F
27$$\"3$)**\\(=YS3M$F;$\"35kRjA0m'R\"F27$$\"3-L$3_ry(GQF;$\"3)Gt[y\"*)
\\m9F27$$\"3h**\\PW@::VF;$\"3BS\"4Rc)eR:F27$$\"3_mm;**pW:[F;$\"3KTiB\"
os&=;F27$$\"3iKLezp5c_F;$\"3!QiSM_\"\\\"p\"F27$$\"3W++]Zd=_dF;$\"3s/*G
_+>vx\"F27$$\"3u****\\i9I]iF;$\"3w_3*)yAIo=F27$$\"3m****\\_#G.t'F;$\"3
=E]exJ<g>F27$$\"3oK$3_cQi;(F;$\"3+V*fZ%)3v/#F27$$\"3'fmm\"*3yXo(F;$\"3
e*4]&[!Qk:#F27$$\"3\"emmmlzO7)F;$\"3/!3#*GEPKD#F27$$\"3E**\\(o;fWj)F;$
\"3chU>&*zJrBF27$$\"3?lmm\"e&e'3*F;$\"3E0\"RM>#*4[#F27$$\"3#)**\\(o\"*
REe*F;$\"3`xk#)zk;2EF27$$\"3$**\\i]4+b+\"F2$\"3!>]Ry#QFLFF27$$\"3om;a8
gya5F2$\"3'f@felg8(GF27$$\"3gmT5se/+6F2$\"3$Q-CX$QI/IF27$$\"3GL$eRuk)[
6F2$\"3w+y['e4Y:$F27$$\"3GL3_JQd*>\"F2$\"377K,&H-(=LF27$$\"3\")*\\78C;
PC\"F2$\"3-qUl\\#z%oMF27$$\"3OL$3KD\"R\"H\"F2$\"3My\">\"*>Wyj$F27$$\"3
&)****\\0UkS8F2$\"38S.#oa/:#QF27$$\"3#)**\\P5'G))Q\"F2$\"3Kjlr_)\\,,%F
27$$\"3!)*\\(o*\\\\aV\"F2$\"3&=jRA<L:?%F27$$\"3%)**\\i[S@([\"F2$\"3'o$
f(**=^ZU%F27$$\"3Ymm;)zEP`\"F2$\"3X,!yg%*>aj%F27$$\"3'****\\i3*Q$e\"F2
$\"3]w$G&)eP9([F27$$\"3,L3_+0RG;F2$\"3WG[0Umm&4&F27$$\"3)****\\F$*)ex;
F2$\"3uL>'z$[j_`F27$$\"3[mTNB3)Qs\"F2$\"3G35#49Vig&F27$$\"3%)*\\Pu=pAx
\"F2$\"3mf.la0>%)eF27$$\"3[mm\"zlx&>=F2$\"3/_ipkBDphF27$$\"3%)*\\(=U_5
p=F2$\"3'fyc=a$\\#['F27$$\"3FLLL#>1o\">F2$\"3iQ!46R3#*z'F27$$\"35L$eMI
(el>F2$\"35$fW(zT5RrF27$$\"3ZmTN#[kR,#F2$\"3=]$edEkH\\(F27$$\"31++]&3=
%e?F2$\"3(>k@)ozcLyF27$$\"3_m;HJpO4@F2$\"3(yPqOB@IC)F27$$\"3'HLLj=O\\:
#F2$\"3EPwA>'Rti)F27$$\"3))*\\(o;D_.AF2$\"3KO^j<.'o0*F27$$\"3#**\\7V$e
-]AF2$\"3Cwrl[4)z[*F27$$\"3#)*************H#F2$\"3A=Z\"[X#=u**F2-%'COL
OURG6&%$RGBGF*F(F(-%*THICKNESSG6#\"\"#-F$6%7dp7$$\"3'******fXqmc\"!#?$
\"3mp:y%GjHQ'!#:7$$\"3\"******>\"4MLJFg[l$\"3#[y!RU;[\">$Fj[l7$$\"3')*
****zO6+q%Fg[l$\"3]B0EGWlF@Fj[l7$$\"3E+++D=omiFg[l$\"3]`2%4#3u&f\"Fj[l
7$$\"3k+++!G_L$yFg[l$\"3!RJcpl#fw7Fj[l7$$\"3;+++PF-+%*Fg[l$\"3y)3<S@FQ
1\"Fj[l7$$\"3)*******=$pm4\"F/$\"3wLB#*z*=&=\"*!#;7$$\"31+++ljL`7F/$\"
3anPq/TqyzFi]l7$$\"30+++5M+59F/$\"3OQqaH9=#4(Fi]l7$$\"39+++c/nm:F/$\"3
_p:y%GjHQ'Fi]l7$$\"3/+++-vLB<F/$\"3QA\"4R*Qp-eFi]l7$$\"3%******za/+)=F
/$\"3V^yQog8>`Fi]l7$$\"3&)*****Rfrm.#F/$\"3RY8ja<(*4\\Fi]l7$$\"3-+++R'
QL>#F/$\"3#>[#)y[f#fXFi]l7$$\"3#******\\o0+N#F/$\"3s\"G5Z&)3`D%Fi]l7$$
\"35+++IFn1DF/$\"3y$)=N_?N*)RFi]l7$$\"3#******\\xRLm#F/$\"3o#eZaG%oaPF
i]l7$$\"3$)*****4#o+?GF/$\"3+Wg^824YNFi]l7$$\"3-+++mQnwHF/$\"3=!zi3Oa%
fLFi]l7$$\"3M+++84MLJF/$\"3?HAPT;[\">$Fi]l7$$\"3#)*****z&z+!H$F/$\"31=
prDj]RIFi]l7$$\"33+++/]nYMF/$\"3>hX&p%pM,HFi]l7$$\"3K+++]?M.OF/$\"3U!p
TsU,_x#Fi]l7$$\"3!)*****\\44+w$F/$\"35c7!\\.o&fEFi]l7$$\"39+++Uhn;RF/$
\"3'p()3EJ&=`DFi]l7$$\"3K+++(=VL2%F/$\"3Mo$=z(e)\\X#Fi]l7$$\"3()*****H
B5+B%F/$\"3;X!R[ZgSO#Fi]l7$$\"3/+++ysn'Q%F/$\"3'4CTRuH'zAFi]l7$$\"3x**
****o8,+ZF/$\"3DKy!yUaw7#Fi]l7$$\"37+++faM8]F/$\"3E:QdEgn%*>Fi]l7$$\"3
;+++xd(fG&F/$\"3EJX7-$)z\"*=Fi]l7$$\"37+++%41'ebF/$\"3%3`fz;7!*z\"Fi]l
7$$\"32+++6kBJeF/$\"3/qgGu@!\\r\"Fi]l7$$\"35+++Hn'Q5'F/$\"3K$**emv0$Q;
Fi]l7$$\"3-+++jt7\\mF/$\"30+#HJvcR]\"Fi]l7$$\"3-+++)*zQ%>(F/$\"3?>'eaH
s**Q\"Fi]l7$$\"3#******pE4\\G)F/$\"3+:j3AQ,27Fi]l7$$\"3#)*****f`IaP*F/
$\"3-Ho[$o<m1\"Fi]l7$$\"3-+++wN=g5F;$\"3on,>F(GBV*F27$$\"3-+++)4CG=\"F
;$\"3+poqkCMa%)F27$$\"3-+++?YY08F;$\"3#*Qfz:!3,m(F27$$\"3))*****>90\"G
9F;$\"3ee\\?]bG-qF27$$\"3'******z17]n\"F;$\"3irCXSi5qfF27$$\"38+++&**=
>#>F;$\"3%*)HCR#H8._F27$$\"3!******f$eQ8CF;$\"3$R2VZEcN9%F27$$\"3))***
***4&Q!pGF;$\"3Mu9jW#)[&[$F27$$\"3%)*****>YS3M$F;$\"3G>CRm(eK*HF27$$\"
3A+++:(y(GQF;$\"3[./b!*))z6EF27$$\"3A+++X@::VF;$\"3i5B3#R:uJ#F27$$\"3#
*********pW:[F;$\"3]')Qs7/lw?F27$$\"3U+++!)p5c_F;$\"3_zNx%z[D!>F27$$\"
3]+++[d=_dF;$\"3!)>h=!fp%Q<F27$$\"3y*****HY,.D'F;$\"3'GE%>HG#**f\"F27$
$\"3q*****HDG.t'F;$\"3UP\\=J;\"e[\"F27$$\"3a+++m&Qi;(F;$\"3?=*=L>KaR\"
F27$$\"3Y+++!4yXo(F;$\"3ckku;wI,8F27$$\"3G+++d'zO7)F;$\"3.Lh*yEp4B\"F2
7$$\"3Q+++n\"fWj)F;$\"3#3K>#H,:e6F27$$\"3m*****>e&e'3*F;$\"3**3m\\$RB0
5\"F27$$\"3#******z\"*REe*F;$\"3,A`koPbV5F27$$\"3$******\\4+b+\"F2$\"3
1x]r0**HX**F;7$$\"31+++9gya5F2$\"39d8?Qff![*F;7$$\"3#******>(e/+6F2$\"
3#\\))z#***H04*F;7$$\"3%******Ruk)[6F2$\"31c#[F#[C/()F;7$$\"3++++KQd*>
\"F2$\"3\"3cVZ)QHO$)F;7$$\"3++++TirV7F2$\"3EMyVX#>//)F;7$$\"3-+++`7R\"
H\"F2$\"3!48$Q)f'eVxF;7$$\"3*)*****f?W1M\"F2$\"3yCZ)*)f+\"fuF;7$$\"3-+
++5'G))Q\"F2$\"3x?FK*\\7.?(F;7$$\"3/++++&\\aV\"F2$\"3e&)=ZQ*ek'pF;7$$
\"35+++\\S@([\"F2$\"3RBRum;)Rs'F;7$$\"3++++)zEP`\"F2$\"3#e%))HLg1?lF;7
$$\"3%******f3*Q$e\"F2$\"37$H1')3nbJ'F;7$$\"3%******4]!RG;F2$\"3L$Ra17
L59'F;7$$\"3++++L*)ex;F2$\"3I8#yE%e$4'fF;7$$\"3,+++B3)Qs\"F2$\"3SH^US]
'3!eF;7$$\"35+++)=pAx\"F2$\"3\\@#3qp$[UcF;7$$\"3/+++ewd>=F2$\"3C$)=mB3
y&\\&F;7$$\"31+++U_5p=F2$\"3)[*4\\DN:]`F;7$$\"3\"******>>1o\">F2$\"3[j
RoF:,<_F;7$$\"3/+++/tel>F2$\"3aNMS[z`(3&F;7$$\"3++++#[kR,#F2$\"3]DzEj3
Ll\\F;7$$\"35+++'3=%e?F2$\"3QvP-h&*4e[F;7$$\"3$)*****4$pO4@F2$\"3REKev
)e2u%F;7$$\"3/+++'=O\\:#F2$\"3LN=uW'30k%F;7$$\"38+++<D_.AF2$\"3;!=^@D)
=QXF;7$$\"37+++Me-]AF2$\"3*p*o!\\T$RWWF;7$Fez$\"3=Al&p3EyM%F;-Fjz6&F\\
[lF(F(F*F][l-F$6$7$7$F*F*7$F*$\"34X!f%G=G=FF2-%&COLORG6&F\\[l$\"\"%!\"
\"FigmFigm-F$6$7$7$$F`[lF)$\"3++++++++]F;7$F`hm$\"3S]1$*)4c!*Q(F2Ffgm-
F$6%7$7$F*F(Fbgm-Fggm6&F\\[l$\"\"&F[hmF\\imF\\im-%*LINESTYLEG6#\"\"$-F
$6%7$7$F`hmF(F_hmFjhmF^im-%)POLYGONSG6<7&7$F*$\"+G=G=F!\"*7$$\"+3VfV5F
]jm$\"+!R/%RGF]jm7$F_jm$\"+pwE#e*!#5Fbgm7&F^jm7$$\"+$[D:3\"F]jm$\"+P]<
\\HF]jm7$Fijm$\"+\\\"*>Y#*FfjmFcjm7&Fhjm7$$\"+cI=C6F]jm$\"+6:qxIF]jm7$
Fb[n$\"+Z%[`*))FfjmF][n7&Fa[n7$$\"+\"RBr;\"F]jm$\"+Vvt7KF]jm7$F[\\n$\"
+gV2o&)FfjmFf[n7&Fj[n7$$\"+Q'f)47F]jm$\"+$)R,`LF]jm7$Fd\\n$\"+8;Ul#)Ff
jmF_\\n7&Fc\\n7$$\"+5;[\\7F]jm$\"+mS`)[$F]jm7$F]]n$\"+v!>L+)FfjmFh\\n7
&F\\]n7$$\"+my]!H\"F]jm$\"+%*>jMOF]jm7$Ff]n$\"+)G())[xFfjmFa]n7&Fe]n7$
$\"+!GPHL\"F]jm$\"+'pl@z$F]jm7$F_^n$\"+k%GA](FfjmFj]n7&F^^n7$$\"+@1Bv8
F]jm$\"+`*))f&RF]jm7$Fh^n$\"+iw]rsFfjmFc^n7&Fg^n7$$\"+AXt=9F]jm$\"+:$)
)=8%F]jm7$Fa_n$\"+[]`[qFfjmF\\_n7&F`_n7$$\"+\"y_qX\"F]jm$\"+0wG$H%F]jm
7$Fj_n$\"+f$pJ'oFfjmFe_n7&Fi_n7$$\"+l+>+:F]jm$\"+k4a#[%F]jm7$Fc`n$\"+S
?#em'FfjmF^`n7&Fb`n7$$\"+vW]V:F]jm$\"+))e'4o%F]jm7$F\\an$\"+0KwykFfjmF
g`n7&F[an7$$\"+NfC&e\"F]jm$\"+::\\!)[F]jm7$Fean$\"+k%p\"3jFfjmF`an7&Fd
an7$$\"+!=^Ji\"F]jm$\"+E'Q!p]F]jm7$F^bn$\"+zb&3;'FfjmFian7&F]bn7$$\"+#
=C#o;F]jm$\"+BFu-`F]jm7$Fgbn$\"+\"=)R%*fFfjmFbbn7&Ffbn7$$\"+FpS1<F]jm$
\"+h684bF]jm7$F`cn$\"+#)oEgeFfjmF[cn7&F_cn7$$\"+OD#3v\"F]jm$\"+$*zLfdF
]jm7$Ficn$\"+V6g6dFfjmFdcn7&Fhcn7$$\"+xy8!z\"F]jm$\"+H$y-*fF]jm7$Fbdn$
\"+**=;'e&FfjmF]dn7&Fadn7$$\"+OIFL=F]jm$\"+BQKaiF]jm7$F[en$\"+d[saaFfj
mFfdn7&Fjdn7$$\"+4zMu=F]jm$\"+P$ol^'F]jm7$Fden$\"+;))=N`FfjmF_en7&Fcen
7$$\"+H_?<>F]jm$\"+n?#>!oF]jm7$F]fn$\"+'oDf@&FfjmFhen7&F\\fn7$$\"+G;cc
>F]jm$\"+b)e\\2(F]jm7$Fffn$\"+5p+6^FfjmFafn7&Fefn7$F`hm$\"+*4c!*Q(F]jm
7$F`hm$\"+++++]FfjmFjfn-Fggm6&F\\[l$\"\"*F[hmFfgnFfgn-%&STYLEG6#%,PATC
HNOGRIDG-%+AXESLABELSG6%Q\"x6\"Q!F`hn-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(
$\"#BF[hm;F($\"\")F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve
 6" "Curve 7" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The graph
 of  " }{XPPEDIT 18 0 "y = 1/x;" "6#/%\"yG*&\"\"\"F&%\"xG!\"\"" }
{TEXT -1 13 " is drawn in " }{TEXT 256 4 "blue" }{TEXT -1 22 ", while \+
the graph of  " }{XPPEDIT 18 0 "y = exp(x);" "6#/%\"yG-%$expG6#%\"xG" 
}{TEXT -1 13 " is drawn in " }{TEXT 260 3 "red" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 56 "We can calculate the area of this region \+
by subtracting " }{XPPEDIT 18 0 "Int(1/x,x = 1 .. 2);" "6#-%$IntG6$*&
\"\"\"F'%\"xG!\"\"/F(;F'\"\"#" }{TEXT -1 7 "  from " }{XPPEDIT 18 0 "I
nt(exp(x),x = 1 .. 2);" "6#-%$IntG6$-%$expG6#%\"xG/F);\"\"\"\"\"#" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }}{PARA 257 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(exp(x),x = 1 .. 2) = exp(x);" "6#
/-%$IntG6$-%$expG6#%\"xG/F*;\"\"\"\"\"#-F(6#F*" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "PIECEWISE([2, ``],[1, ``]);" "6#-%*PIECEWISEG6$7$\"\"#%
!G7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = exp(2)-exp(1);" "6#/%!G,&-%$expG6#\"\"#\"\"\"-F'6#
F*!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "`` = exp(1)*(exp(1)-1);" "6#/%!G*&-%$expG6#\"\"\"F),&-F'6#F)F)F)
!\"\"F)" }{TEXT -1 2 ", " }}{PARA 258 "" 0 "" {TEXT -1 6 "while " }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/x,x = 1 .. 2) =
 ln*x;" "6#/-%$IntG6$*&\"\"\"F(%\"xG!\"\"/F);F(\"\"#*&%#lnGF(F)F(" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2, ``],[1, ``]);" "6#-%*PIEC
EWISEG6$7$\"\"#%!G7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ln*2-ln*1;" "6#/%!G,&*&%#lnG\"\"\"
\"\"#F(F(*&F'F(F(F(!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "`` = ln*2;" "6#/%!G*&%#lnG\"\"\"\"\"#F'" }
{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 26 "so the required area is
:  " }{XPPEDIT 18 0 "exp(1)*(exp(1)-1)-ln*2;" "6#,&*&-%$expG6#\"\"\"F(
,&-F&6#F(F(F(!\"\"F(F(*&%#lnGF(\"\"#F(F," }{TEXT -1 1 " " }{TEXT 272 
1 "~" }{TEXT -1 14 " 3.977627090. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "Int(1/x,x=1..2);\nA1 := valu
e(%);\nInt(exp(x),x=1..2);\nA2 := value(%);\nA2-A1;\nevalf(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'%\"xG!\"\"/F(;F'\"
\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%#lnG6#\"\"#" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$expG6#%\"xG/F);\"\"\"\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G,&-%$expG6#\"\"\"!\"\"-F'6#\"\"#
F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#\"\"\"!\"\"-F%6#\"\"#
F'-%#lnGF*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!4Fw(R!\"*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Alternatively, letting " }
{XPPEDIT 18 0 "f(x) = exp(x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "g(x) = 1/x;" "6#/-%\"gG6#%\"xG*&\"\"\"F)F
'!\"\"" }{TEXT -1 39 ", the required area can be obtained as " }
{XPPEDIT 18 0 "Int(``(f(x)-g(x)),x = 1 .. 2);" "6#-%$IntG6$-%!G6#,&-%
\"fG6#%\"xG\"\"\"-%\"gG6#F-!\"\"/F-;F.\"\"#" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 5 "Now  " }{XPPEDIT 18 0 "f(x)-g(x) = exp(x)-1/x;" 
"6#/,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F(!\"\",&-%$expG6#F(F)*&F)F)F(F-F-" }
{TEXT -1 27 ", so the required area is: " }}{PARA 257 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "Int(``(exp(x)-1/x),x = 1 .. 2) = exp(x)-ln*x;
" "6#/-%$IntG6$-%!G6#,&-%$expG6#%\"xG\"\"\"*&F/F/F.!\"\"F1/F.;F/\"\"#,
&-F,6#F.F/*&%#lnGF/F.F/F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([
2, ``],[1, ``]);" "6#-%*PIECEWISEG6$7$\"\"#%!G7$\"\"\"F(" }{TEXT -1 1 
" " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(exp(2)-
ln*2)-(exp(1)-ln*1);" "6#/%!G,&-F$6#,&-%$expG6#\"\"#\"\"\"*&%#lnGF-F,F
-!\"\"F-,&-F*6#F-F-*&F/F-F-F-F0F0" }{TEXT -1 1 " " }}{PARA 257 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = exp(2)-exp(1)-ln*2;" "6#/%!G,(-%
$expG6#\"\"#\"\"\"-F'6#F*!\"\"*&%#lnGF*F)F*F-" }{TEXT -1 1 " " }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 273 1 "~" }{TEXT -1 16 " 3.977
627090,   " }}{PARA 0 "" 0 "" {TEXT -1 11 "as before. " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Int(exp
(x)-1/x,x=1..2);\nvalue(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%$IntG6$,&-%$expG6#%\"xG\"\"\"*&F+F+F*!\"\"F-/F*;F+\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#\"\"\"!\"\"-F%6#\"\"#F'-%#l
nGF*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!4Fw(R!\"*" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 304 8 "Question" }
{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 59 "Calculate the area of t
he region bounded by the graphs of  " }{XPPEDIT 18 0 "y = x^2-2*x+2;" 
"6#/%\"yG,(*$%\"xG\"\"#\"\"\"*&F(F)F'F)!\"\"F(F)" }{TEXT -1 7 "  and  \+
" }{XPPEDIT 18 0 "y = 5+3*x-x^2;" "6#/%\"yG,(\"\"&\"\"\"*&\"\"$F'%\"xG
F'F'*$F*\"\"#!\"\"" }{TEXT -1 39 " between their points of intersectio
n. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 8 "So
lution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 29 "The two parabo
las meet where " }{XPPEDIT 18 0 "x^2-2*x+2=5+3*x-x^2" "6#/,(*$%\"xG\"
\"#\"\"\"*&F'F(F&F(!\"\"F'F(,(\"\"&F(*&\"\"$F(F&F(F(*$F&F'F*" }{TEXT 
-1 17 ", that is, where " }{XPPEDIT 18 0 "2*x^2-5*x-3=0" "6#/,(*&\"\"#
\"\"\"*$%\"xGF&F'F'*&\"\"&F'F)F'!\"\"\"\"$F,\"\"!" }{TEXT -1 4 " or " 
}{XPPEDIT 18 0 "(2*x+1)*(x-3)=0" "6#/*&,&*&\"\"#\"\"\"%\"xGF(F(F(F(F(,
&F)F(\"\"$!\"\"F(\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
38 "This shows that the graphs meet where " }{XPPEDIT 18 0 "x=-1/2" "6
#/%\"xG,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT -1 11 " and where " }{XPPEDIT 
18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The region can be illustrated a
s follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 561 "f := x -> 5+3*x-x^2:\ng := x -> x^2-2*x+2:\na := -
.5: b := 3:\nc := -1: d := 3.5:\np1 := plot([f(x),g(x)],x=c..d,color=[
red,blue],thickness=2):\np3 := plot([[[a,0],[a,g(a)]],[[b,0],[b,g(b)]]
],\n                color=COLOR(RGB,.5,.5,.5),linestyle=3):\npp := plo
t(f(x),x=a..b,adaptive=false,numpoints=25):\nfpts := op(1,op(1,pp)):\n
pp := plot(g(x),x=a..b,adaptive=false,numpoints=25):\ngpts := op(1,op(
1,pp)):\np4 := plots[polygonplot]([seq([fpts[i-1],fpts[i],gpts[i],gpts
[i-1]],i=2..25)],\n           style=patchnogrid,color=COLOR(RGB,.9,.9,
.9)):\nplots[display]([p1,p3,p4]);" }}{PARA 13 "" 1 "" {GLPLOT2D 318 
399 399 {PLOTDATA 2 "6)-%'CURVESG6%7S7$$!\"\"\"\"!$\"\"\"F*7$$!3K++]i!
G\">!*!#=$\"3w\\A<r[\"3[\"!#<7$$!3!)**\\PMmnl\")F0$\"3!=s:2E9N)=F37$$!
3s***\\PC\")e?(F0$\"3X>Z)=Q))*=BF37$$!3&****\\iqB(RiF0$\"3z<;#)o8uQFF3
7$$!3]+](o9e\"y_F0$\"3NqKq@I'z8$F37$$!3/+]P%yjmQ%F0$\"3#z-;Jns:\\$F37$
$!3c+]P4IdjMF0$\"3$)*yusrk4%QF37$$!3!***\\P47\"*3DF0$\"3;DAi#G!Q%=%F37
$$!3#3+v=-6tb\"F0$\"3]m*[s^a&3XF37$$!3k))***\\iKZy&!#>$\"3(os&43<6B[F3
7$$\"3k&***\\(ovo$GFin$\"3qm$eU[,V3&F37$$\"3O,+]ikFa7F0$\"3!\\=1V%3bg`
F37$$\"3:****\\(o])GAF0$\"3P2$yBXx*=cF37$$\"3E,+]PN.oJF0$\"3%Hv-jpX+&e
F37$$\"3F**\\iS:!4-%F0$\"3?&\\$>qRfWgF37$$\"3@++](3W].&F0$\"3a7$>m`'*p
D'F37$$\"3?+++]e:%*eF0$\"3&z5fJ-O3U'F37$$\"3q**\\ilq]$*oF0$\"3o\"G\\Is
ZGf'F37$$\"3q)****\\A-\"yxF0$\"3p+XFD>WGnF37$$\"33-]i:Lk[()F0$\"3q*>Zg
R0#foF37$$\"3C-](o%z#Gn*F0$\"3ORS;zB@mpF37$$\"3O+]il<rj5F3$\"3-XfawDlf
qF37$$\"3/+D\"GmjA:\"F3$\"3s@%*zR%z!HrF37$$\"3u**\\(o%)yxC\"F3$\"3_%H;
**[%Q'=(F37$$\"3'**\\i!zA*pM\"F3$\"3k7NLP')eEsF37$$\"3;+vVjyNL9F3$\"3!
[QMc#)ebC(F37$$\"3()**\\ig]jE:F3$\"35,bat0H\\sF37$$\"3t****\\K&**Hi\"F
3$\"3'3y/]6r[B(F37$$\"3()**\\7oLF<<F3$\"3L>O]$G#z-sF37$$\"3)**\\i::)[3
=F3$\"3%>bX.1N[:(F37$$\"3#)**\\(ohm(4>F3$\"3a;[p>84#3(F37$$\"3o****\\A
)p2+#F3$\"3O=L([eH#**pF37$$\"3I++vo^$z4#F3$\"3I3>(R`tC*oF37$$\"3y*\\iS
T\")f=#F3$\"3a83c*\\H%znF37$$\"3E++D;#RAG#F3$\"3T&eg&3=5QmF37$$\"3q*\\
ilI5GP#F3$\"3/w#e(o@?)['F37$$\"3%)*\\7G>$[nCF3$\"3PJ`iri(RJ'F37$$\"3/+
+vVK/gDF3$\"3hx*z8K3j7'F37$$\"3!)*\\i!R]%pl#F3$\"3Zr')ew\"y9\"fF37$$\"
3]+++&)HF]FF3$\"3w)=zHY<oo&F37$$\"3/+]P*G9d%GF3$\"31R>P^I0RaF37$$\"3E+
Dc\"Hl.%HF3$\"3Gu?'o#yMv^F37$$\"3x****\\K(Rt-$F3$\"3XHo_TLB<\\F37$$\"3
p**\\(oDAq7$F3$\"3o#QLcd)z-YF37$$\"3W+++&\\zh@$F3$\"35V:%4%zs/VF37$$\"
3m*\\ilqR7J$F3$\"3]*Qr`s5%pRF37$$\"3))*\\P%eWA-MF3$\"3&)f@s4@aJOF37$$
\"3++++++++NF3$\"3+++++++]KF3-%'COLOURG6&%$RGBGF+$F*F*F^[l-%*THICKNESS
G6#\"\"#-F$6%7S7$F($\"\"&F*7$F.$\"36]xdABF<YF37$F5$\"3kxn%e2=**H%F37$F
:$\"3C!GSP\\B/'RF37$F?$\"3X#Q`0E'GPOF37$FD$\"3gH#4O;@UL$F37$FI$\"33skW
[4wpIF37$FN$\"365xy\")zn7GF37$FS$\"31v-W'fGZc#F37$FX$\"3'Q`j0Q9dL#F37$
Fgn$\"3\"RFac'4/>@F37$F]o$\"3.Lmhss1W>F37$Fbo$\"3-:Q%>!o([w\"F37$Fgo$
\"3K#prjh2Rg\"F37$F\\p$\"3iZsWdwvm9F37$Fap$\"3;0!pQ='\\d8F37$Ffp$\"3/(
oI@(y]Y7F37$F[q$\"3G#*3%=cz&o6F37$F`q$\"31=K^$)H]'4\"F37$Feq$\"3i*\\Ds
Ho$\\5F37$Fjq$\"3%)*H:b$*ec,\"F37$F_r$\"3@gM_:/2,5F37$Fdr$\"3*[0z!*=fS
+\"F37$Fir$\"3wyI,BU=B5F37$F^s$\"3a/(epN%Rh5F37$Fcs$\"3x()*G<k./7\"F37
$Fhs$\"3!\\6.y.*z(=\"F37$F]t$\"3W(\\zq[WtF\"F37$Fbt$\"3U=_\\<%G\")Q\"F
37$Fgt$\"3U\"Q@Y3\"[9:F37$F\\u$\"3/[p@\"4`Ol\"F37$Fau$\"3&G=!=(Hvw#=F3
7$Ffu$\"3V!oEwBS:+#F37$F[v$\"3)>4yZjha?#F37$F`v$\"3E'o,X\">b1CF37$Fev$
\"3I:%*o2u8WEF37$Fjv$\"33CU!y83Y)GF37$F_w$\"3Yor=@p]`JF37$Fdw$\"3UA+PA
\\tLMF37$Fiw$\"3wGQZioYXPF37$F^x$\"3s63-AbXjSF37$Fcx$\"3UhI+Q7m1WF37$F
hx$\"3)fU+ZY<]w%F37$F]y$\"3KqJ(4R1,6&F37$Fby$\"3+<;C\"oBU_&F37$Fgy$\"3
Nd%eSb^9\"fF37$F\\z$\"3;56>\")*G=M'F37$Faz$\"3-S`r[BoqnF37$Ffz$\"3++++
+++]sF3-F[[l6&F][lF^[lF^[lF+F_[l-F$6%7$7$$!3++++++++]F0F^[l7$F_elFhz-%
&COLORG6&F][l$Fh[lF)FeelFeel-%*LINESTYLEG6#\"\"$-F$6%7$7$$FielF*F^[l7$
F^flFg[lFbelFfel-%)POLYGONSG6<7&7$$!+++++]!#5$\"++++]K!\"*7$$!+3#*>uMF
gfl$\"+O'Rq$QFjfl7$F\\gl$\"+V/a:GFjflFdfl7&F[gl7$$!+)43m9#Fgfl$\"+2$Q*
4VFjfl7$Fegl$\"+$)3SvCFjflF`gl7&Fdgl7$$!+e/$f`'!#6$\"+[-l*z%Fjfl7$F^hl
$\"+[/*\\8#FjflFigl7&F]hl7$$\"+#z'=$\\)F`hl$\"+#=#eZ_Fjfl7$Fhhl$\"+'o
\\t$=FjflFchl7&Fghl7$$\"+Ft3XBFgfl$\"+_=`[cFjfl7$Fail$\"+!)o(fe\"FjflF
\\il7&F`il7$$\"+Nj&=t$Fgfl$\"+%Q*G!)fFjfl7$Fjil$\"+]i*GR\"FjflFeil7&Fi
il7$$\"+>`xn^Fgfl$\"+yNF$G'Fjfl7$Fcjl$\"+aR]L7FjflF^jl7&Fbjl7$$\"+&y/G
l'Fgfl$\"+?LC`lFjfl7$F\\[m$\"+er.76FjflFgjl7&F[[m7$$\"+W<2L\")Fgfl$\"+
jHXynFjfl7$Fe[m$\"+6U&[.\"FjflF`[m7&Fd[m7$$\"+e#3dl*Fgfl$\"+eaQkpFjfl7
$F^\\m$\"+o`=,5FjflFi[m7&F]\\m7$$\"+LZo*4\"Fjfl$\"+(oZ(*3(Fjfl7$Fg\\m$
\"+Yq$*45FjflFb\\m7&Ff\\m7$$\"+G_m]7Fjfl$\"+r@$y=(Fjfl7$F`]m$\"+cI$G1
\"FjflF[]m7&F_]m7$$\"+jcE-9Fjfl$\"+**zWSsFjfl7$Fi]m$\"+jw\"=;\"FjflFd]
m7&Fh]m7$$\"+t2O[:Fjfl$\"+O7mZsFjfl7$Fb^m$\"+P&*p+8FjflF]^m7&Fa^m7$$\"
+H\"H5o\"Fjfl$\"+b%Gs@(Fjfl7$F[_m$\"+u1!QY\"FjflFf^m7&Fj^m7$$\"+OYyQ=F
jfl$\"+r\\ANrFjfl7$Fd_m$\"+l'fNq\"FjflF__m7&Fc_m7$$\"+VUUs>Fjfl$\"+M`
\"o-(Fjfl7$F]`m$\"+4*3c%>FjflFh_m7&F\\`m7$$\"+x)yy7#Fjfl$\"+;\"od&oFjf
l7$Ff`m$\"+h26sAFjflFa`m7&Fe`m7$$\"+oD[lAFjfl$\"+Qk.kmFjfl7$F_am$\"+Jh
W,EFjflFj`m7&F^am7$$\"+FcX;CFjfl$\"+%34,T'Fjfl7$Fham$\"+VlM1IFjflFcam7
&Fgam7$$\"+\"o<-c#Fjfl$\"+q%Qf7'Fjfl7$Fabm$\"+6#zUV$FjflF\\bm7&F`bm7$$
\"+-$=-r#Fjfl$\"+h;P&y&Fjfl7$Fjbm$\"+Tm%[#RFjflFebm7&Fibm7$$\"+)plz%GF
jfl$\"+y%))HV&Fjfl7$Fccm$\"+?s(\\T%FjflF^cm7&FbcmF_flF_flFgcm-Fcel6&F]
[l$\"\"*F)F]dmF]dm-%&STYLEG6#%,PATCHNOGRIDG-%+AXESLABELSG6%Q\"x6\"Q!Fg
dm-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"#NF)F\\em" 1 2 0 1 10 0 2 9 1 4 
2 1.000000 44.000000 43.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curv
e 4" "Curve 5" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The graph
 of  " }{XPPEDIT 18 0 "y = x^2-2*x+2;" "6#/%\"yG,(*$%\"xG\"\"#\"\"\"*&
F(F)F'F)!\"\"F(F)" }{TEXT -1 13 " is drawn in " }{TEXT 256 4 "blue" }
{TEXT -1 22 ", while the graph of  " }{XPPEDIT 18 0 "y = 5+3*x-x^2;" "
6#/%\"yG,(\"\"&\"\"\"*&\"\"$F'%\"xGF'F'*$F*\"\"#!\"\"" }{TEXT -1 13 " \+
is drawn in " }{TEXT 260 3 "red" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "
f(x) = 5+3*x-x^2;" "6#/-%\"fG6#%\"xG,(\"\"&\"\"\"*&\"\"$F*F'F*F**$F'\"
\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x) = x^2-2*x+2;" "6#/-
%\"gG6#%\"xG,(*$F'\"\"#\"\"\"*&F*F+F'F+!\"\"F*F+" }{TEXT -1 70 ", we c
an find the area of the region enclosed between the curves from " }
{XPPEDIT 18 0 "x = -1/2;" "6#/%\"xG,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT 
-1 4 " to " }{XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 17 " by \+
calculating  " }{XPPEDIT 18 0 "Int(``(f(x)-g(x)),x = -1/2 .. 3);" "6#-
%$IntG6$-%!G6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F-!\"\"/F-;,$*&F.F.\"\"#F2F
2\"\"$" }{TEXT -1 40 ". This will be shorter than calculating " }
{XPPEDIT 18 0 "Int(f(x),x = -1/2 .. 3)-Int(g(x),x = -1/2 .. 3)" "6#,&-
%$IntG6$-%\"fG6#%\"xG/F*;,$*&\"\"\"F/\"\"#!\"\"F1\"\"$F/-F%6$-%\"gG6#F
*/F*;,$*&F/F/F0F1F1F2F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Now  " }{XPPEDIT 18 0 "f(x)-g(x) = \+
3+5*x-2*x^2;" "6#/,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F(!\"\",(\"\"$F)*&\"\"&
F)F(F)F)*&\"\"#F)*$F(F3F)F-" }{TEXT -1 27 ", so the required area is: \+
" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(3+5*x-2*x^
2),x = -1/2 .. 3) = 3*x+5*x^2/2-2*x^3/3;" "6#/-%$IntG6$-%!G6#,(\"\"$\"
\"\"*&\"\"&F,%\"xGF,F,*&\"\"#F,*$F/F1F,!\"\"/F/;,$*&F,F,F1F3F3F+,(*&F+
F,F/F,F,*(F.F,*$F/F1F,F1F3F,*(F1F,*$F/F+F,F+F3F3" }{TEXT -1 2 "  " }
{XPPEDIT 18 0 "PIECEWISE([3, ``],[-1/2, ``]);" "6#-%*PIECEWISEG6$7$\"
\"$%!G7$,$*&\"\"\"F,\"\"#!\"\"F.F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(9+45/2-18)-(-3/2+5/8+1/12);" 
"6#/%!G,&-F$6#,(\"\"*\"\"\"*&\"#XF*\"\"#!\"\"F*\"#=F.F*,(*&\"\"$F*F-F.
F.*&\"\"&F*\"\")F.F**&F*F*\"#7F.F*F." }{TEXT -1 1 " " }}{PARA 257 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -9+48/2-5/8-1/12;" "6#/%!G,*
\"\"*!\"\"*&\"#[\"\"\"\"\"#F'F**&\"\"&F*\"\")F'F'*&F*F*\"#7F'F'" }
{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1
5-17/24" "6#/%!G,&\"#:\"\"\"*&\"#<F'\"#C!\"\"F+" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``=14" "6#/%!G\"#9" }{TEXT -1 1 " " }{XPPEDIT 18 0 "7/2
4 = 343/24;" "6#/*&\"\"(\"\"\"\"#C!\"\"*&\"$V$F&F'F(" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 37 "Int(3+5*x-2*x^2,x=-1/2..3);\nvalue(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$,(\"\"$\"\"\"*&\"\"&F(%\"xGF(F(*&\"\"#F()F+F-F
(!\"\"/F+;#F/F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$V$\"#C" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 274 8 "Q
uestion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 65 "Calculate the
 area of the region enclosed between the graphs of  " }{XPPEDIT 18 0 "
y = sqrt(ln*x);" "6#/%\"yG-%%sqrtG6#*&%#lnG\"\"\"%\"xGF*" }{TEXT -1 7 
"  and  " }{XPPEDIT 18 0 "y = sqrt(ln*x)+sin*x/2;" "6#/%\"yG,&-%%sqrtG
6#*&%#lnG\"\"\"%\"xGF+F+*(%$sinGF+F,F+\"\"#!\"\"F+" }{TEXT -1 6 " from
 " }{XPPEDIT 18 0 "x = Pi/3;" "6#/%\"xG*&%#PiG\"\"\"\"\"$!\"\"" }
{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = Pi;" "6#/%\"xG%#PiG" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 8 "S
olution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 42 "This region c
an be illustrated as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 659 "f := x -> sqrt(ln(x))+sin(x
)/2:\ng := x -> sqrt(ln(x)):\na := Pi/3: b := Pi:\nc := .5: d := 3.3:
\np1 := plot([f(x),g(x)],x=c..d,color=[red,blue],thickness=2):\np2 := \+
plot([[a,g(a)],[a,f(a)]],color=COLOR(RGB,.4,.4,.4)):\np3 := plot([[[a,
0],[a,g(a)]],[[b,0],[b,g(b)]]],\n                color=COLOR(RGB,.5,.5
,.5),linestyle=3):\npp := plot(f(x),x=a..b,adaptive=false,numpoints=25
):\nfpts := op(1,op(1,pp)):\npp := plot(g(x),x=a..b,adaptive=false,num
points=25):\ngpts := op(1,op(1,pp)):\np4 := plots[polygonplot]([seq([f
pts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2..25)],\n           style=patch
nogrid,color=COLOR(RGB,.9,.9,.9)):\nplots[display]([p1,p2,p3,p4],view=
[0..3.3,0..1.3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 510 346 346 {PLOTDATA 
2 "6*-%'CURVESG6%7U7$$\"35+++K&o7+\"!#<$\"3[<KZ'=Ioc%!#=7$$\"3.+++l)=J
+\"F*$\"3'ec[#yLztZF-7$$\"3'******>`>o+\"F*$\"39#\\xj%=3]]F-7$$\"3)***
*****>?0,\"F*$\"3krIib*R&e_F-7$$\"3$******\\`@z,\"F*$\"3sX!4!z['ye&F-7
$$\"36+++qGKD5F*$\"3;urR_lybeF-7$$\"32+++N#\\0/\"F*$\"3RUtfjy62jF-7$$
\"3/++++cxb5F*$\"3]rb`Tj7\"o'F-7$$\"3++++l>+r5F*$\"3mm43SKY2qF-7$$\"3'
*******H$Gi3\"F*$\"3>r(Ge:T.I(F-7$$\"3&******>6^I6\"F*$\"3m()\\Zlg%zv(
F-7$$\"33+++$*Q()R6F*$\"3oY!yzNe6;)F-7$$\"3%******>**p+<\"F*$\"3Mp@TcT
,n&)F-7$$\"3.+++\"4m-?\"F*$\"3$Q%[P+d=L*)F-7$$\"3.+++yheI7F*$\"3?9+A!o
O&o#*F-7$$\"3-+++li!4E\"F*$\"3uoW'3y:nd*F-7$$\"35+++4VM>8F*$\"3c\"fWu4
72,\"F*7$$\"3'******4l6CP\"F*$\"3')Rjq=#QG0\"F*7$$\"34+++aQ^N9F*$\"351
v7d$*p'4\"F*7$$\"3#******ppp*)[\"F*$\"3w'z%*fLm#H6F*7$$\"3-+++^::^:F*$
\"3Oy?)pXsC;\"F*7$$\"3++++FI>1;F*$\"3m'fBxjm!)=\"F*7$$\"3!******4D#em;
F*$\"3k8-'RC$R77F*7$$\"36+++sq3C<F*$\"3yL3jYL<K7F*7$$\"3!******4K(3%y
\"F*$\"3GjF&)e\"H&\\7F*7$$\"3)*******yi=R=F*$\"3]!G*3T6pi7F*7$$\"3!***
****[zh)*=F*$\"3W0WBvl2u7F*7$$\"3++++^2Ng>F*$\"3)Q?)er,)HG\"F*7$$\"3:+
++P$*39?F*$\"3u-x>\")>V)G\"F*7$$\"35+++#[G@2#F*$\"3__,%eQQ?H\"F*7$$\"3
<+++)f)3K@F*$\"3_7dC.,S$H\"F*7$$\"3%)*****H(yu!>#F*$\"3/[$\\xCGDH\"F*7
$$\"3!*******QP]ZAF*$\"3K9#=l]>(*G\"F*7$$\"3)*******\\9_5BF*$\"3gOm%fd
iWG\"F*7$$\"3'******pmXrO#F*$\"3U9Ztn-$zF\"F*7$$\"3!******\\5/wU#F*$\"
3y)>fb!H=p7F*7$$\"3%)*****\\V)Q#[#F*$\"3]!p%Q(*[xf7F*7$$\"3')******y@G
UDF*$\"35A$)*)e!))zC\"F*7$$\"3#******R_P')f#F*$\"3#\\]BqpobB\"F*7$$\"3
%******z4Xvl#F*$\"3&46'37'>8A\"F*7$$\"3@+++=!Q^r#F*$\"3#>FVM,Wi?\"F*7$
$\"33+++pCVvFF*$\"3=FvROhO*=\"F*7$$\"39+++\">.N$GF*$\"3SjaX^^;s6F*7$$
\"37+++\"*)))G*GF*$\"3))ea0j<t`6F*7$$\"37+++[Gy^HF*$\"3M=#)R%4DZ8\"F*7
$$\"3$******zF+f+$F*$\"3lBT')=@s;6F*7$$\"3/+++g\\#z1$F*$\"3]\"Qms3ub4
\"F*7$$\"3z*****\\d+M7$F*$\"3+U)H6e)Hw5F*7$$\"3!******4:\\D=$F*$\"3C`Z
.DS[b5F*7$$\"38+++&)=;RKF*$\"3PTh(H.7a.\"F*7$$\"3#)*************H$F*$
\"3%\\&>yXZz85F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fj[lFi[l-%*THICK
NESSG6#\"\"#-F$6%7P7$F($\"3O[#4r%p^gN!#>7$F/$\"3SL(yB\"GL!e&Fe\\l7$F4$
\"3E61#4a=SC)Fe\\l7$F9$\"3/XyvNz*H-\"F-7$F>$\"3&\\]/s&HxK8F-7$FC$\"3!e
e-0#yP\"e\"F-7$FM$\"3T7RKuPrHBF-7$FW$\"3Sw)\\b!)ff(GF-7$F[o$\"3YEMZ@UD
=OF-7$Feo$\"33'QR[L3DF%F-7$F_p$\"3Mis@s-)[\"[F-7$Fdp$\"3-ru/Z,Ok_F-7$F
ip$\"3Uv2\"[6^ki&F-7$F^q$\"3\"\\f=M?xE,'F-7$Fcq$\"3()*e%f:IR4jF-7$Fhq$
\"3lSP$=L*oDmF-7$F]r$\"3utj;m$)z$)oF-7$Fbr$\"3Ad4ulA&o9(F-7$Fgr$\"3381
xaOO!Q(F-7$F\\s$\"3g9>1#p$f3wF-7$Fas$\"3h?&HS#f\"f!yF-7$Ffs$\"3![>3+#e
.2!)F-7$F[t$\"3S(f?>')4W?)F-7$F`t$\"3YxU$e7*fn$)F-7$Fet$\"3SCah(Goc`)F
-7$Fjt$\"3Kj0z'>h6q)F-7$F_u$\"3!HQl@V\\d&))F-7$Fdu$\"3C#[cLy+!***)F-7$
Fiu$\"3)H!=#[Pc8:*F-7$F^v$\"3I())\\vD)p#G*F-7$Fcv$\"37N0()[Bc<%*F-7$Fh
v$\"3))zj(**)*3``*F-7$F]w$\"3w(>;BM9&f'*F-7$Fbw$\"3!4P$\\`]Ms(*F-7$Fgw
$\"3=JS$))Roj))*F-7$F\\x$\"3Al$pBE7U***F-7$Fax$\"3'=D+9t\\.,\"F*7$Ffx$
\"3gKS^Xea?5F*7$F[y$\"3\\@/WdylI5F*7$F`y$\"3z0bZM!*QS5F*7$Fey$\"3qb6dT
R3\\5F*7$Fjy$\"37x:@ZNxe5F*7$F_z$\"35B;bVI?n5F*7$Fdz$\"3w@f1\\l&f2\"F*
7$Fiz$\"3ch[e#G>T3\"F*7$F^[l$\"3%*yN&GfnE4\"F*-Fc[l6&Fe[lFi[lFi[lFf[lF
[\\l-F$6$7$7$$\"3k(f'>^v>Z5F*$\"3%eT![23]Z@F-7$Fcel$\"3!)3ESEyixkF--%&
COLORG6&Fe[l$\"\"%!\"\"F]flF]fl-F$6%7$7$FcelFi[lFbel-F[fl6&Fe[l$\"\"&F
_flFfflFffl-%*LINESTYLEG6#\"\"$-F$6%7$7$$\"37$z*e`EfTJF*Fi[l7$F`gl$\"3
u/KX/0#*p5F*FdflFhfl-%)POLYGONSG6<7&7$$\"+`v>Z5!\"*$\"+yyixk!#57$$\"+f
7]Q6F\\hl$\"+d\")fT\")F_hl7$Fahl$\"+3^c,OF_hl7$Fjgl$\"+a3]Z@F_hl7&F`hl
7$$\"+DT%z@\"F\\hl$\"+6IFK\"*F_hl7$F]il$\"+8'=.W%F_hlFehl7&F\\il7$$\"+
PfG28F\\hl$\"+KFQ+5F\\hl7$Ffil$\"+d%>k<&F_hlFail7&Feil7$$\"+k*>sR\"F\\
hl$\"+tI$32\"F\\hl7$F_jl$\"+(ojMy&F_hlFjil7&F^jl7$$\"+]ls'[\"F\\hl$\"+
9v(z7\"F\\hl7$Fhjl$\"+cSV(H'F_hlFcjl7&Fgjl7$$\"+81rp:F\\hl$\"+\"4%[r6F
\\hl7$Fa[m$\"+_Q%[r'F_hlF\\[m7&F`[m7$$\"+/ejb;F\\hl$\"+x:E37F\\hl7$Fj[
m$\"+=$*f+rF_hlFe[m7&Fi[m7$$\"+f(*\\W<F\\hl$\"+]MWQ7F\\hl7$Fc\\m$\"+Hs
nfuF_hlF^\\m7&Fb\\m7$$\"+F(yI$=F\\hl$\"+*Qe8E\"F\\hl7$F\\]m$\"+%Q!e%y(
F_hlFg\\m7&F[]m7$$\"+#3$>C>F\\hl$\"+jr6y7F\\hl7$Fe]m$\"+[)e,4)F_hlF`]m
7&Fd]m7$$\"+dmW/?F\\hl$\"+)H1wG\"F\\hl7$F^^m$\"+rI()Q$)F_hlFi]m7&F]^m7
$$\"+tJz%4#F\\hl$\"+T(GGH\"F\\hl7$Fg^m$\"+:f:*f)F_hlFb^m7&Ff^m7$$\"+h1
^&=#F\\hl$\"+j;p#H\"F\\hl7$F`_m$\"+uaAU))F_hlF[_m7&F__m7$$\"+qP$HF#F\\
hl$\"+P;'yG\"F\\hl7$Fi_m$\"+i!*Hh!*F_hlFd_m7&Fh_m7$$\"+HBK_BF\\hl$\"+!
y*zz7F\\hl7$Fb`m$\"+u-!)[#*F_hlF]`m7&Fa`m7$$\"+.IsYCF\\hl$\"+&pagE\"F
\\hl7$F[am$\"+e.7f%*F_hlFf`m7&Fj`m7$$\"+dFpEDF\\hl$\"+u1?^7F\\hl7$Fdam
$\"+^HiF'*F_hlF_am7&Fcam7$$\"+#f;(>EF\\hl$\"+1IhI7F\\hl7$F]bm$\"+KLf8)
*F_hlFham7&F\\bm7$$\"+[%e?q#F\\hl$\"+Q9w47F\\hl7$Ffbm$\"+FW-q**F_hlFab
m7&Febm7$$\"+:0S#z#F\\hl$\"+a4V%=\"F\\hl7$F_cm$\"+Z:O85F\\hlFjbm7&F^cm
7$$\"+FvUyGF\\hl$\"+'y#He6F\\hl7$Fhcm$\"+$yB#G5F\\hlFccm7&Fgcm7$$\"+)o
(=oHF\\hl$\"+(y<$H6F\\hl7$Fadm$\"+l\"\\I/\"F\\hlF\\dm7&F`dm7$$\"+Pbh]I
F\\hl$\"+BS_,6F\\hl7$Fjdm$\"+*=)4c5F\\hlFedm7&Fidm7$$\"+]EfTJF\\hl$\"+
10#*p5F\\hl7$Fcem$\"+/0#*p5F\\hlF^em-F[fl6&Fe[l$\"\"*F_flF\\fmF\\fm-%&
STYLEG6#%,PATCHNOGRIDG-%+AXESLABELSG6%Q\"x6\"Q!Fffm-%%FONTG6#%(DEFAULT
G-%%VIEWG6$;Fi[l$\"#LF_fl;Fi[l$\"#8F_fl" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The \+
graph of  " }{XPPEDIT 18 0 "g(x) = sqrt(ln*x);" "6#/-%\"gG6#%\"xG-%%sq
rtG6#*&%#lnG\"\"\"F'F-" }{TEXT -1 13 " is drawn in " }{TEXT 256 4 "blu
e" }{TEXT -1 22 ", while the graph of  " }{XPPEDIT 18 0 "f(x) = sqrt(l
n*x)+sin*x/2;" "6#/-%\"fG6#%\"xG,&-%%sqrtG6#*&%#lnG\"\"\"F'F.F.*(%$sin
GF.F'F.\"\"#!\"\"F." }{TEXT -1 13 " is drawn in " }{TEXT 260 3 "red" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 89 "There is a problem in trying to find the area enclosed be
tween the graphs by subtracting " }{XPPEDIT 18 0 "Int(g(x),x = Pi/3 ..
 Pi)" "6#-%$IntG6$-%\"gG6#%\"xG/F);*&%#PiG\"\"\"\"\"$!\"\"F-" }{TEXT 
-1 6 " from " }{XPPEDIT 18 0 "Int(f(x),x = Pi/3 .. Pi)" "6#-%$IntG6$-%
\"fG6#%\"xG/F);*&%#PiG\"\"\"\"\"$!\"\"F-" }{TEXT -1 85 ", because it i
s very difficult to find an explicit formula for an anti-derivative of
 " }{XPPEDIT 18 0 "g(x) = sqrt(ln*x);" "6#/-%\"gG6#%\"xG-%%sqrtG6#*&%#
lnG\"\"\"F'F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 68 "We can find the area of the region enclos
ed between the curves from " }{XPPEDIT 18 0 "x = Pi/3;" "6#/%\"xG*&%#P
iG\"\"\"\"\"$!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = Pi;" "6#/%
\"xG%#PiG" }{TEXT -1 16 " by calculating " }{XPPEDIT 18 0 "Int(``(f(x)
-g(x)),x = Pi/3 .. Pi);" "6#-%$IntG6$-%!G6#,&-%\"fG6#%\"xG\"\"\"-%\"gG
6#F-!\"\"/F-;*&%#PiGF.\"\"$F2F6" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 5 "Now  " }{XPPEDIT 18 0 "f(x)-g(x) = sin*x/2;" "6#/,&-%\"fG6
#%\"xG\"\"\"-%\"gG6#F(!\"\"*(%$sinGF)F(F)\"\"#F-" }{TEXT -1 27 ", so t
he required area is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "Int(sin*x/2,x = Pi/3 .. Pi) = -cos*x/2;" "6#/-%$IntG6$*(%$sinG\"
\"\"%\"xGF)\"\"#!\"\"/F*;*&%#PiGF)\"\"$F,F0,$*(%$cosGF)F*F)F+F,F," }
{TEXT -1 2 "  " }{XPPEDIT 18 0 "PIECEWISE([Pi, ``],[Pi/3, ``]);" "6#-%
*PIECEWISEG6$7$%#PiG%!G7$*&F'\"\"\"\"\"$!\"\"F(" }{TEXT -1 1 " " }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(-cos*Pi/2)-(-
cos(Pi/3)/2);" "6#/%!G,&-F$6#,$*(%$cosG\"\"\"%#PiGF+\"\"#!\"\"F.F+,$*&
-F*6#*&F,F+\"\"$F.F+F-F.F.F." }{TEXT -1 1 " " }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2+1/4;" "6#/%!G,&*&\"\"\"F'\"\"#
!\"\"F'*&F'F'\"\"%F)F'" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "`` = 3/4;" "6#/%!G*&\"\"$\"\"\"\"\"%!\"\"" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 35 "Int(sin(x)/2,x=Pi/3..Pi);\nvalue(%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&#\"\"\"\"\"#F)-%$sinG6#%\"xGF)F
)/F.;,$*&\"\"$!\"\"%#PiGF)F)F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"
\"$\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "Calculating the area of a region
 between two graphs .. general case " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Suppose also that \+
" }{XPPEDIT 18 0 "f(x)<=g(x)" "6#1-%\"fG6#%\"xG-%\"gG6#F'" }{TEXT -1 
5 " for " }{XPPEDIT 18 0 "a<=x" "6#1%\"aG%\"xG" }{XPPEDIT 18 0 "``<=b
" "6#1%!G%\"bG" }{TEXT -1 19 " so that the graph " }{XPPEDIT 18 0 "y=f
(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 52 " is above (or at least does
 not go below) the graph " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"gG6#%
\"xG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%
\"aG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 124 "As in the \+
first section, we consider the problem of calculating the area of the \+
region enclosed between the two graphs from " }{XPPEDIT 18 0 "x=a" "6#
/%\"xG%\"aG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "However, this time we \+
" }{TEXT 259 14 "do not require" }{TEXT -1 16 " the two graphs " }
{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT -1 8 " to lie " 
}{TEXT 259 16 "above the x axis" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "a
<=x" "6#1%\"aG%\"xG" }{XPPEDIT 18 0 "``<=b" "6#1%!G%\"bG" }{TEXT -1 2 
". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 557 345 345 
{PLOTDATA 2 "60-%'CURVESG6%7[o7$$\"3++++++++]!#=$\"319@CMUk'y\"!#<7$$
\"3!RL$e*)e&yL&F*$\"3Yhh#H#eI%y\"F-7$$\"3omm;z<rvcF*$\"31Qqg2X*Qy\"F-7
$$\"3!**\\(=#Qy'pfF*$\"3w!*H'f1#4&y\"F-7$$\"37L$3_)\\kjiF*$\"3Wc7#[[7x
y\"F-7$$\"3w+DJv6C%f'F*$\"3o)z_o_9Bz\"F-7$$\"3GnmTlt$[#pF*$\"3aQ%4'*y-
')z\"F-7$$\"3-nm\"zb7/f(F*$\"3p'4$fxA,;=F-7$$\"3%HL3xQCGD)F*$\"3%o5(HK
0()Q=F-7$$\"3%omT&[\\'p'))F*$\"3)y+H^=:S'=F-7$$\"3g+](o#>(G]*F*$\"3yh1
MV$=H*=F-7$$\"3gmTN$y_g,\"F-$\"3&\\#o,[H`C>F-7$$\"3))*\\PHY2;3\"F-$\"3
O\\i[JsJc>F-7$$\"3YLLe3&Q!\\6F-$\"3!\\ncYF,x)>F-7$$\"3pm;H5=V37F-$\"3m
1SPE\\(H,#F-7$$\"37++v+YHv7F-$\"3=mU[>IUP?F-7$$\"37++DO>VU8F-$\"3M,dfP
NTc?F-7$$\"3;++vn:yu8F-$\"3CHnm9)GK1#F-7$$\"3>++D*>JrS\"F-$\"3K/_SaPPo
?F-7$$\"3c$3-Qw2lV\"F-$\"3QbhM@,_r?F-7$$\"3smTNGV)eY\"F-$\"3Y^(eII`J2#
F-7$$\"3&*\\(o\\!f\"3]\"F-$\"3]QACs21t?F-7$$\"3TLLe\"[Zd`\"F-$\"3o=B8#
3.22#F-7$$\"3GL$e*3\"R`c\"F-$\"3m)=MUq1p1#F-7$$\"3OLLLO2$\\f\"F-$\"3+$
4iy-^91#F-7$$\"33+v=J\\xj;F-$\"3)QRAwM&QU?F-7$$\"3ILL$)3PrC<F-$\"3_=*p
!\\))H=?F-7$$\"3E+v=1Kd\"z\"F-$\"3alVZD1j%)>F-7$$\"3;+Dce#R_&=F-$\"3%Q
r\\7@&HY>F-7$$\"3`L$3_5o;#>F-$\"3:V4b@P#3!>F-7$$\"3[L3FB0n#)>F-$\"3m[W
['yK_&=F-7$$\"3ym;Hs)p%[?F-$\"3`SQT,CE.=F-7$$\"3hmTN.p\"o6#F-$\"3/$3qE
+wxu\"F-7$$\"3%)*\\7[>8j<#F-$\"3M2lFt9b*p\"F-7$$\"3$pmT&>3dSAF-$\"3Mt_
e%R1*[;F-7$$\"34++]L_&pI#F-$\"33D5)[;]&*f\"F-7$$\"39+]PJ%**=P#F-$\"3i[
M>?/Zb:F-7$$\"3/+v=#GOZV#F-$\"3n6HwXa#y^\"F-7$$\"3/+]i\"*e]/DF-$\"3On;
A*zzF[\"F-7$$\"3[LL$)))p>nDF-$\"3w2z-SN)zX\"F-7$$\"38++D;J8MEF-$\"3Wrs
.&e\"*)Q9F-7$$\"3K$3-j:gWm#F-$\"3n\"**yM'fyK9F-7$$\"3_mTN'>(y%p#F-$\"3
v2YF)>a#G9F-7$$\"3V$3_:mUzs#F-$\"3B=>K1'o]U\"F-7$$\"3K++vE\")4hFF-$\"3
xXh0'[yOU\"F-7$$\"3#pTNc$[H#z#F-$\"3`)*\\-$)f%RU\"F-7$$\"3^L3_W:\\BGF-
$\"3m2B;BbmD9F-7$$\"3#p;HU4,h&GF-$\"30'z%)[3?*G9F-7$$\"3K+v$Rk5())GF-$
\"3vw[WG6bL9F-7$$\"3CLLeMUZ_HF-$\"3=C@*yv$3Y9F-7$$\"3@+vo/)G#>IF-$\"3K
=NK&H<JY\"F-7$$\"37nmm*Q@N3$F-$\"3[*3B]B-?[\"F-7$$\"33n;zV)p#\\JF-$\"3
N!\\'>e!)Q-:F-7$$\"3vL3_nQZ9KF-$\"3AM/E(o'>A:F-7$$\"3[++]$f*QuKF-$\"3-
'>&z)pu(Q:F-7$$\"3*QLepxfIM$F-$\"35r&et-7Wb\"F-7$$\"3ymmm2#zWS$F-$\"3G
od2Z;<k:F-7$$\"3D$3F%p@APMF-$\"31.\"3*eLRn:F-7$$\"3;+v=J^'*pMF-$\"3KPQ
td82p:F-7$$\"3K++]BVI,NF-$\"3-*f,z%)G\"p:F-7$$\"3Z+D\"e^VE`$F-$\"39w3U
hhen:F-7$$\"31]i!zv@jc$F-$\"3O*>+yYkSc\"F-7$$\"33+++++++OF-$\"3#>(**[+
]ae:F--%'COLOURG6&%$RGBG$\"\"\"\"\"!$Fd_lFd_lFe_l-%*THICKNESSG6#\"\"#-
F$6%7S7$F($!3<%HW8jE9$=F-7$F4$!3+8Mh(\\Hyv\"F-7$F>$!3'\\yddhLJp\"F-7$F
H$!3p(GL!4Mq?;F-7$FM$!3od*Rjip#\\:F-7$FR$!3g'oOJ1#y![\"F-7$FW$!3(oV0QG
X0U\"F-7$Ffn$!3Sf'>OxRBO\"F-7$F[o$!3Hpr`n'=uI\"F-7$F`o$!3gm8SC#H(e7F-7
$Feo$!3))\\')R$=fb@\"F-7$Fjo$!37:]!)[ip$=\"F-7$F_p$!3'[>^o?l\\:\"F-7$F
dp$!3h*4e=&**yL6F-7$F^q$!3))e%RGoX07\"F-7$Fhq$!360+2FWP96F-7$Fbr$!3y65
TZX#Q6\"F-7$F\\s$!3z(\\)G=mh=6F-7$Fas$!3C7d?8e\\H6F-7$Ffs$!3['oS@.GJ9
\"F-7$F[t$!3gHh9ZQYh6F-7$F`t$!3?UoJm(\\7=\"F-7$Fet$!3?[Rf!\\VK?\"F-7$F
jt$!3;,prS$3PA\"F-7$F_u$!33')Q9[q0X7F-7$Fdu$!3Mz:ocrNl7F-7$Fiu$!3#R^De
9X1G\"F-7$F^v$!3%z%emI<*QH\"F-7$Fcv$!3=O[flfH.8F-7$Fhv$!3**fn&=:\"p28F
-7$F]w$!3[.j;$)[+28F-7$Fbw$!3ogn*QKZ-I\"F-7$Fgw$!3%z<I#=/p)G\"F-7$F\\x
$!3g8Z`!*>pq7F-7$Ffx$!39uE%pg\\&\\7F-7$F`y$!3sves**fd@7F-7$Fjy$!3Iz$4k
/K6>\"F-7$Fdz$!3+3%[3G9d:\"F-7$Fiz$!3_9()fO<J=6F-7$F^[l$!3?ax+.7;x5F-7
$Fc[l$!3k-qu$o)eO5F-7$Fh[l$!3k/'R5=%4_**F*7$F]\\l$!3CtQh1uk`&*F*7$Fb\\
l$!3?@<H3w@2#*F*7$Fg\\l$!3h&HXde!4W))F*7$F\\]l$!3!4)*[fvt*e&)F*7$Ff]l$
!3L6_o36.0$)F*7$F`^l$!3(HoxO]4w6)F*7$Fj^l$!3!fN1l.)p$)zF*-F__l6&Fa_lFe
_lFe_lFb_lFf_l-F$6$7$7$$\"3a**************pF*$!3,+++y1a7;F-7$Ffil$\"3$
*******)=f-!=F--%&COLORG6&Fa_l$\"\"%!\"\"F`jlF`jl-F$6$7$7$$\"3!*******
*******R$F-$!31+++V@Ky&)F*7$Fgjl$\"31+++uwhj:F-F]jl-F$6%7$7$Ffil$!33++
+++++@F-Feil-F^jl6&Fa_l$\"\"&FbjlFf[mFf[m-%*LINESTYLEG6#\"\"$-F$6%7$7$
Fgjl$!3/+++++++=F-FfjlFd[mFh[m-%)POLYGONSG6<7&7$$\"+++++q!#5$\"+*=f-!=
!\"*7$$\"+Dj/x\")Fi\\m$\"+e+,O=F\\]m7$F^]m$!+J-X)[\"F\\]m7$Fg\\m$!+z1a
7;F\\]m7&F]]m7$$\"+R!)=,#*Fi\\m$\"+O&>*y=F\\]m7$Fj]m$!+)Hu$*Q\"F\\]mFb
]m7&Fi]m7$$\"+^UHN5F\\]m$\"+9=!R$>F\\]m7$Fc^m$!+W=Z#H\"F\\]mF^^m7&Fb^m
7$$\"+bJB^6F\\]m$\"+H@o))>F\\]m7$F\\_m$!+_uF97F\\]mFg^m7&F[_m7$$\"+A5i
m7F\\]m$\"+-n`M?F\\]m7$Fe_m$!+i8Ee6F\\]mF`_m7&Fd_m7$$\"+Y.gt8F\\]m$\"+
H(3I1#F\\]m7$F^`m$!+5eaE6F\\]mFi_m7&F]`m7$$\"+R7P%[\"F\\]m$\"+fMQt?F\\
]m7$Fg`m$!+Q[`86F\\]mFb`m7&Ff`m7$$\"+b1$*)f\"F\\]m$\"+sjeg?F\\]m7$F`am
$!+*R+\">6F\\]mF[am7&F_am7$$\"+xE78<F\\]m$\"+4IQB?F\\]m7$Fiam$!+bEGS6F
\\]mFdam7&Fham7$$\"+3KeI=F\\]m$\"+V#4='>F\\]m7$Fbbm$!+W9Qt6F\\]mF]bm7&
Fabm7$$\"+3D/M>F\\]m$\"+S]$=*=F\\]m7$F[cm$!+0US27F\\]mFfbm7&Fjbm7$$\"+
vJ^]?F\\]m$\"+$e<;!=F\\]m7$Fdcm$!+EqpX7F\\]mF_cm7&Fccm7$$\"+$3iu;#F\\]
m$\"+FLn1<F\\]m7$F]dm$!+6#R&y7F\\]mFhcm7&F\\dm7$$\"+DS;!G#F\\]m$\"+fy,
>;F\\]m7$Ffdm$!+\"*o1+8F\\]mFadm7&Fedm7$$\"+&=3DQ#F\\]m$\"+iTu[:F\\]m7
$F_em$!+e]#zI\"F\\]mFjdm7&F^em7$$\"+!H0U]#F\\]m$\"+mV\"H[\"F\\]m7$Fhem
$!+=-H+8F\\]mFcem7&Fgem7$$\"+-()H2EF\\]m$\"+_uhX9F\\]m7$Fafm$!+Wqfy7F
\\]mF\\fm7&F`fm7$$\"+[3AFFF\\]m$\"+8'=^U\"F\\]m7$Fjfm$!+k/ZO7F\\]mFefm
7&Fifm7$$\"+nAPLGF\\]m$\"+#>*\\E9F\\]m7$Fcgm$!+hT)f=\"F\\]mF^gm7&Fbgm7
$$\"+)>P)\\HF\\]m$\"+SY[X9F\\]m7$F\\hm$!+`>!*>6F\\]mFggm7&F[hm7$$\"+a$
R21$F\\]m$\"+e#=^Z\"F\\]m7$Fehm$!+Zp+^5F\\]mF`hm7&Fdhm7$$\"+>TXwJF\\]m
$\"+*f\"y5:F\\]m7$F^im$!+%o5Ry*Fi\\mFihm7&F]im7$$\"+&R;FG$F\\]m$\"+i-*
3a\"F\\]m7$Fgim$!+ce/h\"*Fi\\mFbim7&Ffim7$$\"+++++MF\\]m$\"+twhj:F\\]m
7$F`jm$!+R@Ky&)Fi\\mF[jm-F^jl6&Fa_l$\"#&)!\"#FijmFijm-%&STYLEG6#%,PATC
HNOGRIDG-%%TEXTG6%7$$\"#CFbjl$Fi_lFd_lQ)y~=~f(x)6\"F^_l-Fa[n6%7$Fd[n$!
#:FbjlQ)y~=~g(x)Fh[nF`il-Fa[n6%7$$\"\"(Fbjl$!#AFbjlQ$x=aFh[n-F__l6&Fa_
lFd_lFd_lFd_l-Fa[n6%7$$\"#MFbjl$!#>FbjlQ$x=bFh[nFg\\n-%*AXESSTYLEG6#%%
NONEG-%+AXESLABELSG6%Q\"xFh[nQ\"yFh[n-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$F
[[nFbjl$\"$.%F[[n;Fd\\n$\"#@Fbjl" 1 2 0 1 10 0 2 9 1 1 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" }}
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "Both the " }{TEXT 276 1 "
x" }{TEXT -1 5 " and " }{TEXT 277 1 "y" }{TEXT -1 49 " axes can lie an
ywhere in relation to the graphs " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%
\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"
gG6#%\"xG" }{TEXT -1 39 ", and so are omitted from the picture. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Imagine t
hat the region is cut into a large number " }{TEXT 280 1 "n" }{TEXT 
-1 51 " of narrow strips by equally spaced vertical lines " }{XPPEDIT 
18 0 "x=x[i]" "6#/%\"xG&F$6#%\"iG" }{TEXT -1 48 " corresponding to a s
ubdivision of the interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }
{TEXT -1 6 " into " }{TEXT 281 1 "n" }{TEXT -1 14 " subintervals " }
{XPPEDIT 18 0 "[x[i-1],x[i]]" "6#7$&%\"xG6#,&%\"iG\"\"\"F)!\"\"&F%6#F(
" }{TEXT -1 16 " of equal width " }{XPPEDIT 18 0 "h=x[i]-x[i-1]" "6#/%
\"hG,&&%\"xG6#%\"iG\"\"\"&F'6#,&F)F*F*!\"\"F." }{XPPEDIT 18 0 "``=(b-a
)/n" "6#/%!G*&,&%\"bG\"\"\"%\"aG!\"\"F(%\"nGF*" }{TEXT -1 4 " by " }
{TEXT 279 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "a=x[0],x[1],x[2
],` . . . `,x[n]=b" "6'/%\"aG&%\"xG6#\"\"!&F&6#\"\"\"&F&6#\"\"#%(~.~.~
.~G/&F&6#%\"nG%\"bG" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 "
 " }{GLPLOT2D 560 370 370 {PLOTDATA 2 "6M-%'CURVESG6%7[o7$$\"3++++++++
]!#=$\"319@CMUk'y\"!#<7$$\"3!RL$e*)e&yL&F*$\"3Yhh#H#eI%y\"F-7$$\"3omm;
z<rvcF*$\"31Qqg2X*Qy\"F-7$$\"3!**\\(=#Qy'pfF*$\"3w!*H'f1#4&y\"F-7$$\"3
7L$3_)\\kjiF*$\"3Wc7#[[7xy\"F-7$$\"3w+DJv6C%f'F*$\"3o)z_o_9Bz\"F-7$$\"
3GnmTlt$[#pF*$\"3aQ%4'*y-')z\"F-7$$\"3-nm\"zb7/f(F*$\"3p'4$fxA,;=F-7$$
\"3%HL3xQCGD)F*$\"3%o5(HK0()Q=F-7$$\"3%omT&[\\'p'))F*$\"3)y+H^=:S'=F-7
$$\"3g+](o#>(G]*F*$\"3yh1MV$=H*=F-7$$\"3gmTN$y_g,\"F-$\"3&\\#o,[H`C>F-
7$$\"3))*\\PHY2;3\"F-$\"3O\\i[JsJc>F-7$$\"3YLLe3&Q!\\6F-$\"3!\\ncYF,x)
>F-7$$\"3pm;H5=V37F-$\"3m1SPE\\(H,#F-7$$\"37++v+YHv7F-$\"3=mU[>IUP?F-7
$$\"37++DO>VU8F-$\"3M,dfPNTc?F-7$$\"3;++vn:yu8F-$\"3CHnm9)GK1#F-7$$\"3
>++D*>JrS\"F-$\"3K/_SaPPo?F-7$$\"3c$3-Qw2lV\"F-$\"3QbhM@,_r?F-7$$\"3sm
TNGV)eY\"F-$\"3Y^(eII`J2#F-7$$\"3&*\\(o\\!f\"3]\"F-$\"3]QACs21t?F-7$$
\"3TLLe\"[Zd`\"F-$\"3o=B8#3.22#F-7$$\"3GL$e*3\"R`c\"F-$\"3m)=MUq1p1#F-
7$$\"3OLLLO2$\\f\"F-$\"3+$4iy-^91#F-7$$\"33+v=J\\xj;F-$\"3)QRAwM&QU?F-
7$$\"3ILL$)3PrC<F-$\"3_=*p!\\))H=?F-7$$\"3E+v=1Kd\"z\"F-$\"3alVZD1j%)>
F-7$$\"3;+Dce#R_&=F-$\"3%Qr\\7@&HY>F-7$$\"3`L$3_5o;#>F-$\"3:V4b@P#3!>F
-7$$\"3[L3FB0n#)>F-$\"3m[W['yK_&=F-7$$\"3ym;Hs)p%[?F-$\"3`SQT,CE.=F-7$
$\"3hmTN.p\"o6#F-$\"3/$3qE+wxu\"F-7$$\"3%)*\\7[>8j<#F-$\"3M2lFt9b*p\"F
-7$$\"3$pmT&>3dSAF-$\"3Mt_e%R1*[;F-7$$\"34++]L_&pI#F-$\"33D5)[;]&*f\"F
-7$$\"39+]PJ%**=P#F-$\"3i[M>?/Zb:F-7$$\"3/+v=#GOZV#F-$\"3n6HwXa#y^\"F-
7$$\"3/+]i\"*e]/DF-$\"3On;A*zzF[\"F-7$$\"3[LL$)))p>nDF-$\"3w2z-SN)zX\"
F-7$$\"38++D;J8MEF-$\"3Wrs.&e\"*)Q9F-7$$\"3K$3-j:gWm#F-$\"3n\"**yM'fyK
9F-7$$\"3_mTN'>(y%p#F-$\"3v2YF)>a#G9F-7$$\"3V$3_:mUzs#F-$\"3B=>K1'o]U
\"F-7$$\"3K++vE\")4hFF-$\"3xXh0'[yOU\"F-7$$\"3#pTNc$[H#z#F-$\"3`)*\\-$
)f%RU\"F-7$$\"3^L3_W:\\BGF-$\"3m2B;BbmD9F-7$$\"3#p;HU4,h&GF-$\"30'z%)[
3?*G9F-7$$\"3K+v$Rk5())GF-$\"3vw[WG6bL9F-7$$\"3CLLeMUZ_HF-$\"3=C@*yv$3
Y9F-7$$\"3@+vo/)G#>IF-$\"3K=NK&H<JY\"F-7$$\"37nmm*Q@N3$F-$\"3[*3B]B-?[
\"F-7$$\"33n;zV)p#\\JF-$\"3N!\\'>e!)Q-:F-7$$\"3vL3_nQZ9KF-$\"3AM/E(o'>
A:F-7$$\"3[++]$f*QuKF-$\"3-'>&z)pu(Q:F-7$$\"3*QLepxfIM$F-$\"35r&et-7Wb
\"F-7$$\"3ymmm2#zWS$F-$\"3God2Z;<k:F-7$$\"3D$3F%p@APMF-$\"31.\"3*eLRn:
F-7$$\"3;+v=J^'*pMF-$\"3KPQtd82p:F-7$$\"3K++]BVI,NF-$\"3-*f,z%)G\"p:F-
7$$\"3Z+D\"e^VE`$F-$\"39w3Uhhen:F-7$$\"31]i!zv@jc$F-$\"3O*>+yYkSc\"F-7
$$\"33+++++++OF-$\"3#>(**[+]ae:F--%'COLOURG6&%$RGBG$\"\"\"\"\"!$Fd_lFd
_lFe_l-%*THICKNESSG6#\"\"#-F$6%7S7$F($!3<%HW8jE9$=F-7$F4$!3+8Mh(\\Hyv
\"F-7$F>$!3'\\yddhLJp\"F-7$FH$!3p(GL!4Mq?;F-7$FM$!3od*Rjip#\\:F-7$FR$!
3g'oOJ1#y![\"F-7$FW$!3(oV0QGX0U\"F-7$Ffn$!3Sf'>OxRBO\"F-7$F[o$!3Hpr`n'
=uI\"F-7$F`o$!3gm8SC#H(e7F-7$Feo$!3))\\')R$=fb@\"F-7$Fjo$!37:]!)[ip$=
\"F-7$F_p$!3'[>^o?l\\:\"F-7$Fdp$!3h*4e=&**yL6F-7$F^q$!3))e%RGoX07\"F-7
$Fhq$!360+2FWP96F-7$Fbr$!3y65TZX#Q6\"F-7$F\\s$!3z(\\)G=mh=6F-7$Fas$!3C
7d?8e\\H6F-7$Ffs$!3['oS@.GJ9\"F-7$F[t$!3gHh9ZQYh6F-7$F`t$!3?UoJm(\\7=
\"F-7$Fet$!3?[Rf!\\VK?\"F-7$Fjt$!3;,prS$3PA\"F-7$F_u$!33')Q9[q0X7F-7$F
du$!3Mz:ocrNl7F-7$Fiu$!3#R^De9X1G\"F-7$F^v$!3%z%emI<*QH\"F-7$Fcv$!3=O[
flfH.8F-7$Fhv$!3**fn&=:\"p28F-7$F]w$!3[.j;$)[+28F-7$Fbw$!3ogn*QKZ-I\"F
-7$Fgw$!3%z<I#=/p)G\"F-7$F\\x$!3g8Z`!*>pq7F-7$Ffx$!39uE%pg\\&\\7F-7$F`
y$!3sves**fd@7F-7$Fjy$!3Iz$4k/K6>\"F-7$Fdz$!3+3%[3G9d:\"F-7$Fiz$!3_9()
fO<J=6F-7$F^[l$!3?ax+.7;x5F-7$Fc[l$!3k-qu$o)eO5F-7$Fh[l$!3k/'R5=%4_**F
*7$F]\\l$!3CtQh1uk`&*F*7$Fb\\l$!3?@<H3w@2#*F*7$Fg\\l$!3h&HXde!4W))F*7$
F\\]l$!3!4)*[fvt*e&)F*7$Ff]l$!3L6_o36.0$)F*7$F`^l$!3(HoxO]4w6)F*7$Fj^l
$!3!fN1l.)p$)zF*-F__l6&Fa_lFe_lFe_lFb_lFf_l-F$6$7$7$$\"3a*************
*pF*$!3,+++y1a7;F-7$Ffil$\"3$*******)=f-!=F--%&COLORG6&Fa_l$\"\"%!\"\"
F`jlF`jl-F$6$7$7$$\"3M+++++++zF*$!3'*******G,(o^\"F-7$Fgjl$\"32+++G@2E
=F-F]jl-F$6$7$7$$\"3-+++++++))F*$!3$******fDNpU\"F-7$Fb[m$\"3'******p&
37h=F-F]jl-F$6$7$7$$\"3u*************p*F*$!3)******Hhz_M\"F-7$F]\\m$\"
3-+++-gF->F-F]jl-F$6$7$7$$\"31++++++g5F-$!3'*******462u7F-7$Fh\\m$\"3+
+++&3/f%>F-F]jl-F$6$7$7$$\"3#*************\\6F-$!30+++sm*\\@\"F-7$Fc]m
$\"3++++!QJ\"))>F-F]jl-F$6$7$7$$\"3**************R7F-$!35+++_5=p6F-7$F
^^m$\"3-+++T_7D?F-F]jl-F$6$7$7$$\"32++++++I8F-$!3'******\\(Q8P6F-7$Fi^
m$\"3=+++2DP`?F-F]jl-F$6$7$7$$\"3$*************>9F-$!35+++-ws=6F-7$Fd_
m$\"3))******)zM*p?F-F]jl-F$6$7$7$$\"3,++++++5:F-$!36+++\"[2K6\"F-7$F_
`m$\"3#)*****>CiE2#F-F]jl-F$6$7$7$$\"33+++++++;F-$!3.+++/IB>6F-7$Fj`m$
\"35+++\"4].1#F-F]jl-F$6$7$7$$\"3%**************o\"F-$!3#******Hw[\\8
\"F-7$Feam$\"3&******H$p#G.#F-F]jl-F$6$7$7$$\"3-++++++!y\"F-$!3!******
f7\"3e6F-7$F`bm$\"3-+++h!o4*>F-F]jl-F$6$7$7$$\"35++++++q=F-$!3)******
\\&e0'=\"F-7$F[cm$\"3,+++kIjO>F-F]jl-F$6$7$7$$\"3'*************f>F-$!3
%******>sIh@\"F-7$Ffcm$\"35+++P@`s=F-F]jl-F$6$7$7$$\"3#)************\\
?F-$!3/+++dk`X7F-7$Fadm$\"3/+++33.-=F-F]jl-F$6$7$7$$\"38++++++S@F-$!3#
******H))=;F\"F-7$F\\em$\"3,+++Ms!*G<F-F]jl-F$6$7$7$$\"3)*************
HAF-$!3/+++)3v>H\"F-7$Fgem$\"3-+++M'zql\"F-F]jl-F$6$7$7$$\"3%)********
****>BF-$!3%*******pnd/8F-7$Fbfm$\"3++++HOK!f\"F-F]jl-F$6$7$7$$\"39+++
+++5CF-$!3********4X(yI\"F-7$F]gm$\"3%******z%***>`\"F-F]jl-F$6$7$7$$
\"3++++++++DF-$!3!******ftx3I\"F-7$Fhgm$\"3)*******[A\"[[\"F-F]jl-F$6$
7$7$$\"3')*************e#F-$!31+++Aj?$G\"F-7$Fchm$\"31+++$e71X\"F-F]jl
-F$6$7$7$$\"3;++++++!o#F-$!3%******\\C6^D\"F-7$F^im$\"3(******Hho-V\"F
-F]jl-F$6$7$7$$\"3-++++++qFF-$!3/+++@KY<7F-7$Fiim$\"3%******R=-OU\"F-F
]jl-F$6$7$7$$\"3))************fGF-$!3-+++9*4<<\"F-7$Fdjm$\"34+++MPSH9F
-F]jl-F$6$7$7$$\"3=++++++]HF-$!3&******H(Q!)>6F-7$F_[n$\"3%******RW@bW
\"F-F]jl-F$6$7$7$$\"3/++++++SIF-$!3/+++<[5k5F-7$Fj[n$\"37+++C(=!p9F-F]
jl-F$6$7$7$$\"3!*************HJF-$!3%******H>hs+\"F-7$Fe\\n$\"33+++!e)
Q'\\\"F-F]jl-F$6$7$7$$\"3>++++++?KF-$!3%)******4*p2_*F*7$F`]n$\"3)****
***[$4Q_\"F-F]jl-F$6$7$7$$\"30++++++5LF-$!39++++^q8!*F*7$F[^n$\"3-+++m
KUZ:F-F]jl-F$6$7$7$$\"3!**************R$F-$!31+++V@Ky&)F*7$Ff^n$\"31++
+uwhj:F-F]jl-F$6%7$7$Ffil$!33+++++++@F-Feil-F^jl6&Fa_l$\"\"&FbjlFe_nFe
_n-%*LINESTYLEG6#\"\"$-F$6%7$7$Ff^n$!3/+++++++=F-Fe^nFc_nFg_n-%)POLYGO
NSG6<7&7$$\"+++++q!#5$\"+*=f-!=!\"*7$$\"+Dj/x\")Fh`n$\"+e+,O=F[an7$F]a
n$!+J-X)[\"F[an7$Ff`n$!+z1a7;F[an7&F\\an7$$\"+R!)=,#*Fh`n$\"+O&>*y=F[a
n7$Fian$!+)Hu$*Q\"F[anFaan7&Fhan7$$\"+^UHN5F[an$\"+9=!R$>F[an7$Fbbn$!+
W=Z#H\"F[anF]bn7&Fabn7$$\"+bJB^6F[an$\"+H@o))>F[an7$F[cn$!+_uF97F[anFf
bn7&Fjbn7$$\"+A5im7F[an$\"+-n`M?F[an7$Fdcn$!+i8Ee6F[anF_cn7&Fccn7$$\"+
Y.gt8F[an$\"+H(3I1#F[an7$F]dn$!+5eaE6F[anFhcn7&F\\dn7$$\"+R7P%[\"F[an$
\"+fMQt?F[an7$Ffdn$!+Q[`86F[anFadn7&Fedn7$$\"+b1$*)f\"F[an$\"+sjeg?F[a
n7$F_en$!+*R+\">6F[anFjdn7&F^en7$$\"+xE78<F[an$\"+4IQB?F[an7$Fhen$!+bE
GS6F[anFcen7&Fgen7$$\"+3KeI=F[an$\"+V#4='>F[an7$Fafn$!+W9Qt6F[anF\\fn7
&F`fn7$$\"+3D/M>F[an$\"+S]$=*=F[an7$Fjfn$!+0US27F[anFefn7&Fifn7$$\"+vJ
^]?F[an$\"+$e<;!=F[an7$Fcgn$!+EqpX7F[anF^gn7&Fbgn7$$\"+$3iu;#F[an$\"+F
Ln1<F[an7$F\\hn$!+6#R&y7F[anFggn7&F[hn7$$\"+DS;!G#F[an$\"+fy,>;F[an7$F
ehn$!+\"*o1+8F[anF`hn7&Fdhn7$$\"+&=3DQ#F[an$\"+iTu[:F[an7$F^in$!+e]#zI
\"F[anFihn7&F]in7$$\"+!H0U]#F[an$\"+mV\"H[\"F[an7$Fgin$!+=-H+8F[anFbin
7&Ffin7$$\"+-()H2EF[an$\"+_uhX9F[an7$F`jn$!+Wqfy7F[anF[jn7&F_jn7$$\"+[
3AFFF[an$\"+8'=^U\"F[an7$Fijn$!+k/ZO7F[anFdjn7&Fhjn7$$\"+nAPLGF[an$\"+
#>*\\E9F[an7$Fb[o$!+hT)f=\"F[anF][o7&Fa[o7$$\"+)>P)\\HF[an$\"+SY[X9F[a
n7$F[\\o$!+`>!*>6F[anFf[o7&Fj[o7$$\"+a$R21$F[an$\"+e#=^Z\"F[an7$Fd\\o$
!+Zp+^5F[anF_\\o7&Fc\\o7$$\"+>TXwJF[an$\"+*f\"y5:F[an7$F]]o$!+%o5Ry*Fh
`nFh\\o7&F\\]o7$$\"+&R;FG$F[an$\"+i-*3a\"F[an7$Ff]o$!+ce/h\"*Fh`nFa]o7
&Fe]o7$$\"+++++MF[an$\"+twhj:F[an7$F_^o$!+R@Ky&)Fh`nFj]o-F^jl6&Fa_l$\"
#&)!\"#Fh^oFh^o-%&STYLEG6#%,PATCHNOGRIDG-%%TEXTG6%7$$\"#CFbjl$Fi_lFd_l
Q)y~=~f(x)6\"F^_l-F`_o6%7$Fc_o$!#:FbjlQ)y~=~g(x)Fg_oF`il-F`_o6%7$$\"\"
(Fbjl$!#AFbjlQ$x=aFg_o-F__l6&Fa_lFd_lFd_lFd_l-F`_o6%7$$\"#MFbjl$!#>Fbj
lQ$x=bFg_oFf`o-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q\"xFg_oQ\"yFg_o-%
%FONTG6#%(DEFAULTG-%%VIEWG6$;$Fj^oFbjl$\"$.%Fj^o;Fc`o$\"#@Fbjl" 1 2 0 
1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" 
"Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" 
"Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Cur
ve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 2
2" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "
Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curv
e 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40" }}{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "Pick a value " }{XPPEDIT 18 0 
"x=x*`*`[i]" "6#/%\"xG*&F$\"\"\"&%\"*G6#%\"iGF&" }{TEXT -1 38 "  in ea
ch of the associated intervals " }{XPPEDIT 18 0 "[x[i-1], x[i]]" "6#7$
&%\"xG6#,&%\"iG\"\"\"F)!\"\"&F%6#F(" }{TEXT -1 8 ", where " }{XPPEDIT 
18 0 "i=1,2,` . . . `,n" "6&/%\"iG\"\"\"\"\"#%(~.~.~.~G%\"nG" }{TEXT 
-1 15 ". For example, " }{XPPEDIT 18 0 "x*`*`[i]" "6#*&%\"xG\"\"\"&%\"
*G6#%\"iGF%" }{TEXT -1 24 " could be the mid-point " }{XPPEDIT 18 0 "(
x[i]+x[i-1])/2;" "6#*&,&&%\"xG6#%\"iG\"\"\"&F&6#,&F(F)F)!\"\"F)F)\"\"#
F-" }{TEXT -1 107 " of each subinterval. Then we can replace each of t
he strips above by a genuine rectangular strip of width " }{XPPEDIT 
18 0 "Delta*x = h;" "6#/*&%&DeltaG\"\"\"%\"xGF&%\"hG" }{TEXT -1 12 " a
nd height " }{XPPEDIT 18 0 "f(x*`*`[i])-g(x*`*`[i])" "6#,&-%\"fG6#*&%
\"xG\"\"\"&%\"*G6#%\"iGF)F)-%\"gG6#*&F(F)&F+6#F-F)!\"\"" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 16 "The area of the " }{TEXT 259 11 "re
ctangular" }{TEXT -1 33 " strip associated with the index " }{TEXT 
282 1 "i" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "(f(x*`*`[i])-g(x*`*`[i]))
*`.`*Delta*x;" "6#**,&-%\"fG6#*&%\"xG\"\"\"&%\"*G6#%\"iGF*F*-%\"gG6#*&
F)F*&F,6#F.F*!\"\"F*%\".GF*%&DeltaGF*F)F*" }{TEXT -1 2 ". " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 
568 369 369 {PLOTDATA 2 "6cq-%'CURVESG6%7[o7$$\"3++++++++]!#=$\"319@CM
Uk'y\"!#<7$$\"3!RL$e*)e&yL&F*$\"3Yhh#H#eI%y\"F-7$$\"3omm;z<rvcF*$\"31Q
qg2X*Qy\"F-7$$\"3!**\\(=#Qy'pfF*$\"3w!*H'f1#4&y\"F-7$$\"37L$3_)\\kjiF*
$\"3Wc7#[[7xy\"F-7$$\"3w+DJv6C%f'F*$\"3o)z_o_9Bz\"F-7$$\"3GnmTlt$[#pF*
$\"3aQ%4'*y-')z\"F-7$$\"3-nm\"zb7/f(F*$\"3p'4$fxA,;=F-7$$\"3%HL3xQCGD)
F*$\"3%o5(HK0()Q=F-7$$\"3%omT&[\\'p'))F*$\"3)y+H^=:S'=F-7$$\"3g+](o#>(
G]*F*$\"3yh1MV$=H*=F-7$$\"3gmTN$y_g,\"F-$\"3&\\#o,[H`C>F-7$$\"3))*\\PH
Y2;3\"F-$\"3O\\i[JsJc>F-7$$\"3YLLe3&Q!\\6F-$\"3!\\ncYF,x)>F-7$$\"3pm;H
5=V37F-$\"3m1SPE\\(H,#F-7$$\"37++v+YHv7F-$\"3=mU[>IUP?F-7$$\"37++DO>VU
8F-$\"3M,dfPNTc?F-7$$\"3;++vn:yu8F-$\"3CHnm9)GK1#F-7$$\"3>++D*>JrS\"F-
$\"3K/_SaPPo?F-7$$\"3c$3-Qw2lV\"F-$\"3QbhM@,_r?F-7$$\"3smTNGV)eY\"F-$
\"3Y^(eII`J2#F-7$$\"3&*\\(o\\!f\"3]\"F-$\"3]QACs21t?F-7$$\"3TLLe\"[Zd`
\"F-$\"3o=B8#3.22#F-7$$\"3GL$e*3\"R`c\"F-$\"3m)=MUq1p1#F-7$$\"3OLLLO2$
\\f\"F-$\"3+$4iy-^91#F-7$$\"33+v=J\\xj;F-$\"3)QRAwM&QU?F-7$$\"3ILL$)3P
rC<F-$\"3_=*p!\\))H=?F-7$$\"3E+v=1Kd\"z\"F-$\"3alVZD1j%)>F-7$$\"3;+Dce
#R_&=F-$\"3%Qr\\7@&HY>F-7$$\"3`L$3_5o;#>F-$\"3:V4b@P#3!>F-7$$\"3[L3FB0
n#)>F-$\"3m[W['yK_&=F-7$$\"3ym;Hs)p%[?F-$\"3`SQT,CE.=F-7$$\"3hmTN.p\"o
6#F-$\"3/$3qE+wxu\"F-7$$\"3%)*\\7[>8j<#F-$\"3M2lFt9b*p\"F-7$$\"3$pmT&>
3dSAF-$\"3Mt_e%R1*[;F-7$$\"34++]L_&pI#F-$\"33D5)[;]&*f\"F-7$$\"39+]PJ%
**=P#F-$\"3i[M>?/Zb:F-7$$\"3/+v=#GOZV#F-$\"3n6HwXa#y^\"F-7$$\"3/+]i\"*
e]/DF-$\"3On;A*zzF[\"F-7$$\"3[LL$)))p>nDF-$\"3w2z-SN)zX\"F-7$$\"38++D;
J8MEF-$\"3Wrs.&e\"*)Q9F-7$$\"3K$3-j:gWm#F-$\"3n\"**yM'fyK9F-7$$\"3_mTN
'>(y%p#F-$\"3v2YF)>a#G9F-7$$\"3V$3_:mUzs#F-$\"3B=>K1'o]U\"F-7$$\"3K++v
E\")4hFF-$\"3xXh0'[yOU\"F-7$$\"3#pTNc$[H#z#F-$\"3`)*\\-$)f%RU\"F-7$$\"
3^L3_W:\\BGF-$\"3m2B;BbmD9F-7$$\"3#p;HU4,h&GF-$\"30'z%)[3?*G9F-7$$\"3K
+v$Rk5())GF-$\"3vw[WG6bL9F-7$$\"3CLLeMUZ_HF-$\"3=C@*yv$3Y9F-7$$\"3@+vo
/)G#>IF-$\"3K=NK&H<JY\"F-7$$\"37nmm*Q@N3$F-$\"3[*3B]B-?[\"F-7$$\"33n;z
V)p#\\JF-$\"3N!\\'>e!)Q-:F-7$$\"3vL3_nQZ9KF-$\"3AM/E(o'>A:F-7$$\"3[++]
$f*QuKF-$\"3-'>&z)pu(Q:F-7$$\"3*QLepxfIM$F-$\"35r&et-7Wb\"F-7$$\"3ymmm
2#zWS$F-$\"3God2Z;<k:F-7$$\"3D$3F%p@APMF-$\"31.\"3*eLRn:F-7$$\"3;+v=J^
'*pMF-$\"3KPQtd82p:F-7$$\"3K++]BVI,NF-$\"3-*f,z%)G\"p:F-7$$\"3Z+D\"e^V
E`$F-$\"39w3Uhhen:F-7$$\"31]i!zv@jc$F-$\"3O*>+yYkSc\"F-7$$\"33+++++++O
F-$\"3#>(**[+]ae:F--%'COLOURG6&%$RGBG$\"\"\"\"\"!$Fd_lFd_lFe_l-%*THICK
NESSG6#Fc_l-F$6%7S7$F($!3<%HW8jE9$=F-7$F4$!3+8Mh(\\Hyv\"F-7$F>$!3'\\yd
dhLJp\"F-7$FH$!3p(GL!4Mq?;F-7$FM$!3od*Rjip#\\:F-7$FR$!3g'oOJ1#y![\"F-7
$FW$!3(oV0QGX0U\"F-7$Ffn$!3Sf'>OxRBO\"F-7$F[o$!3Hpr`n'=uI\"F-7$F`o$!3g
m8SC#H(e7F-7$Feo$!3))\\')R$=fb@\"F-7$Fjo$!37:]!)[ip$=\"F-7$F_p$!3'[>^o
?l\\:\"F-7$Fdp$!3h*4e=&**yL6F-7$F^q$!3))e%RGoX07\"F-7$Fhq$!360+2FWP96F
-7$Fbr$!3y65TZX#Q6\"F-7$F\\s$!3z(\\)G=mh=6F-7$Fas$!3C7d?8e\\H6F-7$Ffs$
!3['oS@.GJ9\"F-7$F[t$!3gHh9ZQYh6F-7$F`t$!3?UoJm(\\7=\"F-7$Fet$!3?[Rf!
\\VK?\"F-7$Fjt$!3;,prS$3PA\"F-7$F_u$!33')Q9[q0X7F-7$Fdu$!3Mz:ocrNl7F-7
$Fiu$!3#R^De9X1G\"F-7$F^v$!3%z%emI<*QH\"F-7$Fcv$!3=O[flfH.8F-7$Fhv$!3*
*fn&=:\"p28F-7$F]w$!3[.j;$)[+28F-7$Fbw$!3ogn*QKZ-I\"F-7$Fgw$!3%z<I#=/p
)G\"F-7$F\\x$!3g8Z`!*>pq7F-7$Ffx$!39uE%pg\\&\\7F-7$F`y$!3sves**fd@7F-7
$Fjy$!3Iz$4k/K6>\"F-7$Fdz$!3+3%[3G9d:\"F-7$Fiz$!3_9()fO<J=6F-7$F^[l$!3
?ax+.7;x5F-7$Fc[l$!3k-qu$o)eO5F-7$Fh[l$!3k/'R5=%4_**F*7$F]\\l$!3CtQh1u
k`&*F*7$Fb\\l$!3?@<H3w@2#*F*7$Fg\\l$!3h&HXde!4W))F*7$F\\]l$!3!4)*[fvt*
e&)F*7$Ff]l$!3L6_o36.0$)F*7$F`^l$!3(HoxO]4w6)F*7$Fj^l$!3!fN1l.)p$)zF*-
F__l6&Fa_lFe_lFe_lFb_lFf_l-F$6%7$7$$\"3a**************pF*$!33+++++++@F
-7$Feil$!33+++'zeTc\"F--%&COLORG6&Fa_l$\"\"&!\"\"F_jlF_jl-%*LINESTYLEG
6#\"\"$-F$6%7$7$$\"3!**************R$F-$!3/+++++++=F-7$Fjjl$!37+++(ofa
y)F*F\\jlFbjl-F$6$7$Fiil7$Feil$\"3!******H$=%=\"=F--F]jl6&Fa_l$\"\"%Fa
jlFi[mFi[m-F$6$7$7$$\"3M+++++++zF*Fjil7$F_\\m$\"3&******R!HiU=F-Fg[m-F
$6$7$7$$\"3-+++++++))F*$!3'******RL>5Z\"F-7$Fh\\m$\"3!******HO`6)=F-Fg
[m-F$6$7$7$$\"3u*************p*F*$!3'******>*o#\\Q\"F-7$Fc]m$\"3%*****
*Rm?S#>F-Fg[m-F$6$7$7$$\"31++++++g5F-$!33+++#o^#38F-7$F^^m$\"3/+++hmVn
>F-Fg[m-F$6$7$7$$\"3#*************\\6F-$!35+++$*G$HC\"F-7$Fi^m$\"3=+++
.F^2?F-Fg[m-F$6$7$7$$\"3**************R7F-$!3++++G[Q!>\"F-7$Fd_m$\"3z*
*****o?aS?F-Fg[m-F$6$7$7$$\"32++++++I8F-$!35+++1%H9:\"F-7$F_`m$\"33+++
p#pK1#F-Fg[m-F$6$7$7$$\"3$*************>9F-$!31+++L%ei7\"F-7$Fj`m$\"3=
+++8n7t?F-Fg[m-F$6$7$7$$\"3,++++++5:F-$!3$*******[>)[6\"F-7$FeamF_amFg
[m-F$6$7$7$$\"33+++++++;F-$!3++++%>8g7\"F-7$F^bm$\"39+++y:Uo?F-Fg[m-F$
6$7$7$$\"3%**************o\"F-$!3-+++5KuX6F-7$Fibm$\"35+++Z$f%[?F-Fg[m
-F$6$7$7$$\"3-++++++!y\"F-$!3))*****\\FK;<\"F-7$Fdcm$\"3'*******o]f8?F
-Fg[m-F$6$7$7$$\"35++++++q=F-$!3*******R*[+,7F-7$F_dm$\"31+++4'3_'>F-F
g[m-F$6$7$7$$\"3'*************f>F-$!3-+++cu3J7F-7$Fjdm$\"3)******Hq0c!
>F-Fg[m-F$6$7$7$$\"3#)************\\?F-$!3#******zn]\"f7F-7$Feem$\"34+
++j>&y$=F-Fg[m-F$6$7$7$$\"38++++++S@F-$!3!******4k\\EG\"F-7$F`fm$\"35+
++$)=bl<F-Fg[m-F$6$7$7$$\"3)*************HAF-$!3)******RPa$*H\"F-7$F[g
m$\"3++++F))e#p\"F-Fg[m-F$6$7$7$$\"3%)************>BF-$!31+++$RkuI\"F-
7$Ffgm$\"35+++#)[%Gi\"F-Fg[m-F$6$7$7$$\"39++++++5CF-Fhgm7$Fahm$\"3%***
***z\"3#*f:F-Fg[m-F$6$7$7$$\"3++++++++DF-$!3,++++9q08F-7$Fjhm$\"3!****
**>]wo]\"F-Fg[m-F$6$7$7$$\"3')*************e#F-$!3!******z[vLH\"F-7$Fe
im$\"3!******p_5gY\"F-Fg[m-F$6$7$7$$\"3;++++++!o#F-$!3!******\\\\@/F\"
F-7$F`jm$\"3%******\\0&pQ9F-Fg[m-F$6$7$7$$\"3-++++++qFF-$!34+++&o.uB\"
F-7$F[[n$\"3.+++?yFD9F-Fg[m-F$6$7$7$$\"3))************fGF-$!3)******\\
\\)[&>\"F-7$Ff[n$\"3++++\"[SjV\"F-Fg[m-F$6$7$7$$\"3=++++++]HF-$!3/+++X
nQY6F-7$Fa\\n$\"3++++$HclX\"F-Fg[m-F$6$7$7$$\"3/++++++SIF-$!3'******f?
oA4\"F-7$F\\]n$\"3#*******fUX#[\"F-Fg[m-F$6$7$7$$\"3!*************HJF-
$!3%*******QGlN5F-7$Fg]n$\"3*******p>O.^\"F-Fg[m-F$6$7$7$$\"3>++++++?K
F-$!33+++n<%Gz*F*7$Fb^n$\"32+++BpKO:F-Fg[m-F$6$7$7$$\"30++++++5LF-$!3a
******[C\"*f#*F*7$F]_n$\"3)******Hrcmb\"F-Fg[m-F$6$7$F^[m7$FjjlFb_nFg[
m-F$6$7$FiilF^\\mFg[m-F$6$7$7$F_\\mFj\\mFg\\mFg[m-F$6$7$7$Fh\\mFe]mFb]
mFg[m-F$6$7$7$Fc]mF`^mF]^mFg[m-F$6$7$7$F^^mF[_mFh^mFg[m-F$6$7$7$Fi^mFf
_mFc_mFg[m-F$6$7$7$Fd_mFa`mF^`mFg[m-F$6$7$7$F_`mF\\amFi`mFg[m-F$6$7$7$
Fj`m$!3++++?vU96F-7$FeamF[bnFg[m-F$6$7$Fdam7$F^bmFgamFg[m-F$6$7$F]bm7$
FibmF`bmFg[m-F$6$7$Fhbm7$FdcmF[cmFg[m-F$6$7$Fccm7$F_dmFfcmFg[m-F$6$7$F
^dm7$FjdmFadmFg[m-F$6$7$Fidm7$FeemF\\emFg[m-F$6$7$Fdem7$F`fmFgemFg[m-F
$6$7$F_fm7$F[gmFbfmFg[m-F$6$7$Fjfm7$FfgmF]gmFg[m-F$6$7$FegmF`hmFg[m-F$
6$7$7$FahmF\\imFihmFg[m-F$6$7$7$FjhmFgimFdimFg[m-F$6$7$7$FeimFbjmF_jmF
g[m-F$6$7$7$F`jmF][nFjjmFg[m-F$6$7$7$F[[nFh[nFe[nFg[m-F$6$7$7$Ff[nFc\\
nF`\\nFg[m-F$6$7$7$Fa\\nF^]nF[]nFg[m-F$6$7$7$F\\]nFi]nFf]nFg[m-F$6$7$7
$Fg]nFd^nFa^nFg[m-F$6$7$7$Fb^nF__nF\\_nFg[m-F$6$7$7$F]_nF_[mF^[mFg[m-F
$6$7$Fd[m7$F_\\mFe[mFg[m-F$6$7$Fa\\m7$Fh\\mFb\\mFg[m-F$6$7$F\\]m7$Fc]m
F]]mFg[m-F$6$7$Fg]m7$F^^mFh]mFg[m-F$6$7$Fb^m7$Fi^mFc^mFg[m-F$6$7$F]_m7
$Fd_mF^_mFg[m-F$6$7$Fh_m7$F_`mFi_mFg[m-F$6$7$Fc`m7$Fj`mFd`mFg[m-F$6$7$
F^amFiamFg[m-F$6$7$7$FeamFcbmFbbmFg[m-F$6$7$7$F^bmF^cmF]cmFg[m-F$6$7$7
$FibmFicmFhcmFg[m-F$6$7$7$FdcmFddmFcdmFg[m-F$6$7$7$F_dmF_emF^emFg[m-F$
6$7$7$FjdmFjemFiemFg[m-F$6$7$7$FeemFefmFdfmFg[m-F$6$7$7$F`fmF`gmF_gmFg
[m-F$6$7$7$F[gmF[hmFjgmFg[m-F$6$7$7$FfgmFdhmFchmFg[m-F$6$7$7$FahmF_imF
^imFg[m-F$6$7$7$FjhmFjimFiimFg[m-F$6$7$7$FeimFejmFdjmFg[m-F$6$7$7$F`jm
F`[nF_[nFg[m-F$6$7$7$F[[n$\"3*)*****>Se]U\"F-7$Ff[nF`]oFg[m-F$6$7$Fj[n
7$Fa\\nF[\\nFg[m-F$6$7$Fe\\n7$F\\]nFf\\nFg[m-F$6$7$F`]n7$Fg]nFa]nFg[m-
F$6$7$F[^n7$Fb^nF\\^nFg[m-F$6$7$Ff^n7$F]_nFg^nFg[m-F$6$7$Fa_nFg_nFg[m-
%)POLYGONSG6B7&7$$\"\"(Fajl$\"+L=%=\"=!\"*7$$\"+++++z!#5Fa_o7$Fe_o$!+'
zeTc\"Fc_o7$F__oFi_o7&7$Fe_o$\"+/HiU=Fc_o7$$\"+++++))Fg_oF^`o7$Fa`o$!+
M$>5Z\"Fc_o7$Fe_oFd`o7&7$Fa`o$\"+jL:\")=Fc_o7$$\"+++++(*Fg_oFi`o7$F\\a
o$!+#*o#\\Q\"Fc_o7$Fa`oF_ao7&7$F\\ao$\"+k1-C>Fc_o7$$\"++++g5Fc_oFdao7$
Fgao$!+#o^#38Fc_o7$F\\aoFjao7&7$Fgao$\"+hmVn>Fc_o7$$\"++++]6Fc_oF_bo7$
Fbbo$!+$*G$HC\"Fc_o7$FgaoFebo7&7$Fbbo$\"+.F^2?Fc_o7$$\"++++S7Fc_oFjbo7
$F]co$!+G[Q!>\"Fc_o7$FbboF`co7&7$F]co$\"+p?aS?Fc_o7$$\"++++I8Fc_oFeco7
$Fhco$!+1%H9:\"Fc_o7$F]coF[do7&7$Fhco$\"+p#pK1#Fc_o7$$\"++++?9Fc_oF`do
7$Fcdo$!+L%ei7\"Fc_o7$FhcoFfdo7&7$Fcdo$\"+8n7t?Fc_o7$$\"++++5:Fc_oF[eo
7$F^eo$!+?vU96Fc_o7$FcdoFaeo7&7$F^eo$\"+y:Uo?Fc_o7$$\"+++++;Fc_oFfeo7$
Fieo$!+\\>)[6\"Fc_o7$F^eoF\\fo7&7$Fieo$\"+Z$f%[?Fc_o7$$\"++++!p\"Fc_oF
afo7$Fdfo$!+%>8g7\"Fc_o7$FieoFgfo7&7$Fdfo$\"+p]f8?Fc_o7$$\"++++!y\"Fc_
oF\\go7$F_go$!+5KuX6Fc_o7$FdfoFbgo7&7$F_go$\"+4'3_'>Fc_o7$$\"++++q=Fc_
oFggo7$Fjgo$!+vAjr6Fc_o7$F_goF]ho7&7$Fjgo$\"+.dg0>Fc_o7$$\"++++g>Fc_oF
bho7$Feho$!+%*[+,7Fc_o7$FjgoFhho7&7$Feho$\"+j>&y$=Fc_o7$$\"++++]?Fc_oF
]io7$F`io$!+cu3J7Fc_o7$FehoFcio7&7$F`io$\"+$)=bl<Fc_o7$$\"++++S@Fc_oFh
io7$F[jo$!+y1:f7Fc_o7$F`ioF^jo7&7$F[jo$\"+F))e#p\"Fc_o7$$\"++++IAFc_oF
cjo7$Ffjo$!+T'\\EG\"Fc_o7$F[joFijo7&7$Ffjo$\"+#)[%Gi\"Fc_o7$$\"++++?BF
c_oF^[p7$Fa[p$!+uVN*H\"Fc_o7$FfjoFd[p7&7$Fa[p$\"+=3#*f:Fc_o7$$\"++++5C
Fc_oFi[p7$F\\\\p$!+$RkuI\"Fc_o7$Fa[pF_\\p7&7$F\\\\p$\"+-l(o]\"Fc_o7$$
\"+++++DFc_oFd\\p7$Fg\\p$!++9q08Fc_o7$F\\\\pFj\\p7&7$Fg\\p$\"+F0,m9Fc_
o7$$\"++++!f#Fc_oF_]p7$Fb]p$!+)[vLH\"Fc_o7$Fg\\pFe]p7&7$Fb]p$\"+b]pQ9F
c_o7$$\"++++!o#Fc_oFj]p7$F]^p$!+&\\@/F\"Fc_o7$Fb]pF`^p7&7$F]^p$\"+?yFD
9Fc_o7$$\"++++qFFc_oFe^p7$Fh^p$!+&o.uB\"Fc_o7$F]^pF[_p7&7$Fh^p$\"+-%e]
U\"Fc_o7$$\"++++gGFc_oF`_p7$Fc_p$!+&\\)[&>\"Fc_o7$Fh^pFf_p7&7$Fc_p$\"+
\"[SjV\"Fc_o7$$\"++++]HFc_oF[`p7$F^`p$!+XnQY6Fc_o7$Fc_pFa`p7&7$F^`p$\"
+$HclX\"Fc_o7$$\"++++SIFc_oFf`p7$Fi`p$!+1#oA4\"Fc_o7$F^`pF\\ap7&7$Fi`p
$\"+gUX#[\"Fc_o7$$\"++++IJFc_oFaap7$Fdap$!+RGlN5Fc_o7$Fi`pFgap7&7$Fdap
$\"+(>O.^\"Fc_o7$$\"++++?KFc_oF\\bp7$F_bp$!+n<%Gz*Fg_o7$FdapFbbp7&7$F_
bp$\"+BpKO:Fc_o7$$\"++++5LFc_oFgbp7$Fjbp$!+\\C\"*f#*Fg_o7$F_bpF]cp7&7$
Fjbp$\"+8nlc:Fc_o7$$\"+++++MFc_oFbcp7$Fecp$!+(ofay)Fg_o7$FjbpFhcp-F]jl
6&Fa_l$\"#&)!\"#F]dpF]dp-%&STYLEG6#%,PATCHNOGRIDG-%%TEXTG6%7$$\"#CFajl
$\"\"#Fd_lQ)y~=~f(x)6\"F^_l-Fedp6%7$Fhdp$!#:FajlQ)y~=~g(x)F]epF_il-Fed
p6%7$F__o$!#AFajlQ$x=aF]ep-F__l6&Fa_lFd_lFd_lFd_l-Fedp6%7$$\"#MFajl$!#
>FajlQ$x=bF]epFjep-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q\"xF]epQ\"yF]
ep-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$F_dpFajl$\"$.%F_dp;Fgep$\"#@Fajl" 1 
2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve
 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve
 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" 
"Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Cur
ve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 2
8" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "
Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curv
e 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47
" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "C
urve 54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curve 59" "Curve
 60" "Curve 61" "Curve 62" "Curve 63" "Curve 64" "Curve 65" "Curve 66
" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Curve 72" "C
urve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "Curve 78" "Curve
 79" "Curve 80" "Curve 81" "Curve 82" "Curve 83" "Curve 84" "Curve 85
" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90" "Curve 91" "C
urve 92" "Curve 93" "Curve 94" "Curve 95" "Curve 96" "Curve 97" "Curve
 98" "Curve 99" "Curve 100" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The total area of the region en
closed between the two graphs from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"
aG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 
29 " is approximated by the sum: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "Sum((f(x*`*`[i])-g(x*`*`[i]))*`.`*Delta*x,i = 1 .. n);
" "6#-%$SumG6$**,&-%\"fG6#*&%\"xG\"\"\"&%\"*G6#%\"iGF-F--%\"gG6#*&F,F-
&F/6#F1F-!\"\"F-%\".GF-%&DeltaGF-F,F-/F1;F-%\"nG" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 32 "Taking the limit of this sum as " }
{XPPEDIT 18 0 "n->infinity" "6#f*6#%\"nG7\"6$%)operatorG%&arrowG6\"%)i
nfinityGF*F*F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Delta;" "6#%&Delta
G" }{XPPEDIT 18 0 "x->0" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"!F
*F*F*" }{TEXT -1 67 " gives the area of the region enclosed between th
e two graphs from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 4 "
 to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "On the other hand,
 by a similar but simpler argument applied to the single function " }
{XPPEDIT 18 0 "h(x)=f(x)-g(x)" "6#/-%\"hG6#%\"xG,&-%\"fG6#F'\"\"\"-%\"
gG6#F'!\"\"" }{TEXT -1 16 ", we know that: " }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(Sum(h(x*`*`[i])*`.`*Delta*x,i = 1
 .. n),n = infinity) = Int(h(x),x = a .. b);" "6#/-%&LimitG6$-%$SumG6$
**-%\"hG6#*&%\"xG\"\"\"&%\"*G6#%\"iGF0F0%\".GF0%&DeltaGF0F/F0/F4;F0%\"
nG/F9%)infinityG-%$IntG6$-F,6#F//F/;%\"aG%\"bG" }{TEXT -1 2 ", " }}
{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "Limit(Sum((f(x*`*`[i])-g(x*`*`[i]))*`.`*Delta*x,i \+
= 1 .. n),n = infinity) = Int(``(f(x)-g(x)),x = a .. b);" "6#/-%&Limit
G6$-%$SumG6$**,&-%\"fG6#*&%\"xG\"\"\"&%\"*G6#%\"iGF1F1-%\"gG6#*&F0F1&F
36#F5F1!\"\"F1%\".GF1%&DeltaGF1F0F1/F5;F1%\"nG/FA%)infinityG-%$IntG6$-
%!G6#,&-F-6#F0F1-F76#F0F</F0;%\"aG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 66 "Hence the area of the region enclosed between the \+
two graphs from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 4 " t
o " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 27 " is given by th
e integral: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(`
`(f(x)-g(x)),x = a .. b)" "6#-%$IntG6$-%!G6#,&-%\"fG6#%\"xG\"\"\"-%\"g
G6#F-!\"\"/F-;%\"aG%\"bG" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT 
-1 1 " " }{TEXT 283 11 "___________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "As in the case of the a
rea of a region under a graph, we can simplfy the notation and write: \+
" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(Sum((f(x)-g
(x))*`.`*Delta*x,x = a .. b),Delta*x = 0) = Int(``(f(x)-g(x)),x = a ..
 b);" "6#/-%&LimitG6$-%$SumG6$**,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F/!\"\"F0
%\".GF0%&DeltaGF0F/F0/F/;%\"aG%\"bG/*&%&DeltaGF0F/F0\"\"!-%$IntG6$-%!G
6#,&-F-6#F/F0-F26#F/F4/F/;F9F:" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 18 "where we think of " }{TEXT 286 1 "x" }{TEXT -1 65 " as ru
nning through a large number of values starting at or near " }{TEXT 
284 1 "a" }{TEXT -1 24 ", and ending at or near " }{TEXT 285 1 "b" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 64 "The following schematic picture may be used in this conne
ction. " }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 525 383 383 
{PLOTDATA 2 "6>-%'CURVESG6%7eo7$$\"35+++++++?!#=$\"3W:@-1,&[)=!#<7$$\"
3ILLLe=HGGF*$\"3'*3ZG**H]X=F-7$$\"3ymmTvT)*[NF*$\"3$4_Fp_#3==F-7$$\"31
+]7=6BaRF*$\"3>e@CR2$f!=F-7$$\"3MLL$31y%fVF*$\"39O\\5!)[L'z\"F-7$$\"3w
LLLV7TnZF*$\"37RFb!fs$*y\"F-7$$\"3kLL$eUW`<&F*$\"3W.*G%e)*=&y\"F-7$$\"
37m\"z%*fT$y`F*$\"3G#)R-S]:%y\"F-7$$\"3s**\\7t(Q8e&F*$\"3%)\\m;0g\"Qy
\"F-7$$\"3IL3xYfL%y&F*$\"3eYI15&pTy\"F-7$$\"3)om;/7Lt)fF*$\"3%p;?&y&4_
y\"F-7$$\"3#****\\7&=ujjF*$\"3OHad\"[B*)y\"F-7$$\"3&HL$3#e],u'F*$\"3_s
sakm)[z\"F-7$$\"3d***\\(e%\\'>vF*$\"3(z&>7D)yQ\"=F-7$$\"3sLL37$3eK)F*$
\"3e>i*35!oT=F-7$$\"3`)**\\P!=QH\"*F*$\"37INF\")zlv=F-7$$\"3Qnmm6f&f&*
*F*$\"3L<.UL')f9>F-7$$\"3GLLe$G+%o5F-$\"3atwb!)\\'*\\>F-7$$\"3)*****\\
B6O]6F-$\"3382]mFH))>F-7$$\"3()****\\-&eEB\"F-$\"3]1c&\\%3RA?F-7$$\"3*
)****\\ws'>J\"F-$\"3!e!HXN[c[?F-7$$\"3_mT&)es(zM\"F-$\"3-:sPHfpd?F-7$$
\"3;L$37C()RQ\"F-$\"3Ex7rzs'[1#F-7$$\"3.+v=$f1oU\"F-$\"3+K%>$pdkq?F-7$
$\"3omm;Xfip9F-$\"3+hCOH,Dt?F-7$$\"3]m;/\\Fw([\"F-$\"35WCvuxNt?F-7$$\"
3bmm\"Hb**e]\"F-$\"3w'e#R#ffG2#F-7$$\"3gm;zcj.C:F-$\"3'fb'\\+'[<2#F-7$
$\"3lmmmgJ<U:F-$\"339yQP&>+2#F-7$$\"3;L3x*)zO%e\"F-$\"3+NZqd))ej?F-7$$
\"3()**\\()=GcE;F-$\"3/Q!*[8&*y`?F-7$$\"3ammml>E,<F-$\"3)yW-%*eW$G?F-7
$$\"3')**\\()o(=Ky\"F-$\"3%\\<g;4E#*)>F-7$$\"3/+]iE5Eh=F-$\"34kRCDiQU>
F-7$$\"3ommTN**oU>F-$\"3AR#zkHpa)=F-7$$\"3rm;/$4nu,#F-$\"3K:J8`<-G=F-7
$$\"3?LLefV7)4#F-$\"3A)Q8PI2Iw\"F-7$$\"3CL$37f/>=#F-$\"3Wwj3sl1&p\"F-7
$$\"3=+]7Hb$[D#F-$\"3v87*f2**zj\"F-7$$\"3NLL3SHgLBF-$\"3Q8/!)HB*3e\"F-
7$$\"34+++$Qx\\T#F-$\"3gK%>gBy!H:F-7$$\"3u***\\(*R'e%\\#F-$\"3Y4*z]W(H
([\"F-7$$\"3=+](o@7;d#F-$\"3Y&\\<\"*z'[c9F-7$$\"3$**\\ibBuVh#F-$\"3dS]
4-KsV9F-7$$\"35++Dai8dEF-$\"3&z$)*>xb6M9F-7$$\"3UL$e*H)fbp#F-$\"3$3Rg[
4f\"G9F-7$$\"3Jmmm0M)Rt#F-$\"37XIpNMoC9F-7$$\"3]**\\()*3'\\aFF-$\"3A&3
N?I:QU\"F-7$$\"39LL3u(3]x#F-$\"3)pW.8K8OU\"F-7$$\"3wm;He9_&z#F-$\"3<N&
*o$*y0C9F-7$$\"3&*****\\UT.;GF-$\"3`Z7]mo7D9F-7$$\"3'o;aQA4K&GF-$\"38a
/$\\1u&G9F-7$$\"3KL$3_I%Q!*GF-$\"3aG\"\\8kBQV\"F-7$$\"3E++]r'o;(HF-$\"
35rH%y6H1X\"F-7$$\"3cm;aD4:[IF-$\"3S_wC$z$Rr9F-7$$\"3E+]P=p4GJF-$\"3Y*
Hv;e&z&\\\"F-7$$\"3fmm;R(ei?$F-$\"3dp=pOXx>:F-7$$\"3u**\\(=#p3)G$F-$\"
3@$[l$QeAU:F-7$$\"3aLLL(=(*oO$F-$\"3!4Zq;(Rte:F-7$$\"3vL$epi%>2MF-$\"3
EgQGrX\\k:F-7$$\"3_LLem?\\ZMF-$\"3)G)\\ag84o:F-7$$\"3#o\"H#[%RZnMF-$\"
3ywi)Ha+!p:F-7$$\"39+D1BeX([$F-$\"3S*euy3'Hp:F-7$$\"3V$3-8qPu]$F-$\"3)
R/vaFb*o:F-7$$\"3um;az&>u_$F-$\"3;Edg4w&zc\"F-7$$\"3uL3F@@9kNF-$\"3m)y
Gm@_Vc\"F-7$$\"3I+++jY'3g$F-$\"3in/(*4lPe:F-7$$\"3IL$e*RG&Hk$F-$\"3ewN
\"4cJ&[:F-7$$\"3umm\"p,T]o$F-$\"3gU;r(pSa`\"F-7$$\"37LLL^$H.w$F-$\"3Eh
\"*)>3iR]\"F-7$$\"34+]()=CgSQF-$\"3u4J-*)QYf9F-7$$\"3!***\\7()RV<RF-$
\"32NG))*37tS\"F-7$$\"\"%\"\"!$\"3]wI!p:_DM\"F--%'COLOURG6&%$RGBG$\"*+
+++\"!\")$F^blF^blFhbl-%*THICKNESSG6#\"\"#-F$6%7Y7$F($!3?ld#QAi&>@F-7$
F/$!3Onb>9H7\\?F-7$F4$!3St!)y+!o8)>F-7$F>$!3EE-\"*\\$)R**=F-7$FH$!3,a`
,L#pC\"=F-7$Ffn$!3PWRT;)eNs\"F-7$F`o$!3;z%fjJP3k\"F-7$Feo$!3iVCjw#fnb
\"F-7$Fjo$!3MyA'R;UMZ\"F-7$F_p$!3U%*y:)emfR\"F-7$Fdp$!3Ve]N!*=)QK\"F-7
$Fip$!3W?GT/A-o7F-7$F^q$!3Dsi_$z&y97F-7$Fcq$!3E-FW_`Ss6F-7$Fhq$!3*3([J
7*[C9\"F-7$Fbr$!3XU%yJ@!\\C6F-7$Fgr$!3F>#3T5ty6\"F-7$F\\s$!3$R4dg$H;96
F-7$Ffs$!3K<,`ed>86F-7$F`t$!3.M>;L=796F-7$Fjt$!3%='4n*\\eH7\"F-7$F_u$!
3logU\"*y[P6F-7$Fdu$!3]PiN@T,f6F-7$Fiu$!3]LK'>$G?$=\"F-7$F^v$!3PnLwHNJ
57F-7$Fcv$!3]e(*RSb:N7F-7$Fhv$!3)\\\"R(e%o0g7F-7$F]w$!3!\\*4mrD%>G\"F-
7$Fbw$!3))zyD\"e-jH\"F-7$Fgw$!3-4&4:6.dI\"F-7$F\\x$!3AlI(ywixI\"F-7$Fa
x$!3)))H7Oe*f,8F-7$Ffx$!3+Fzz_9o(G\"F-7$F`y$!3!Gbb;j)>j7F-7$Fjy$!3?K'R
JY&fL7F-7$F^[l$!39y%z;Nh\\>\"F-7$Fh[l$!3Y[OsURwa6F-7$F]\\l$!3&z*>Q3Sk1
6F-7$Fb\\l$!3I!zn6ei*e5F-7$Fg\\l$!3SAN3,gX35F-7$F\\]l$!3;z9\"*[=#Gg*F*
7$Fa]l$!3XR$yT\\b:8*F*7$Ff]l$!3kz+x9#>&G()F*7$F`^l$!3[y:w%G^eQ)F*7$Fd_
l$!3`$e?%*\\&*48)F*7$Fi_l$!3'eGJ4w8f/)F*7$F^`l$!3g\\w())QfC)zF*7$Fc`l$
!3`N!\\p;Ku$zF*7$Fh`l$!3`=)o2^mH#zF*7$$\"3%***\\7%=&oAPF-$!3)H1\"*=oGl
$zF*7$F]al$!3OIM+QlXvzF*7$$\"3$o;/^)eY+QF-$!3[\"*)fGeU]/)F*7$Fbal$!3ZS
#*R*y:N9)F*7$Fgal$!3djA4,)G6T)F*7$F\\bl$!3`^?04pM4))F*-Fbbl6&FdblFhblF
hblFeblFibl-F$6%7$7$$\"3a**************pF*$!33+++++++@F-7$F_^m$!3,+++y
1a7;F--Fbbl6&FdblF^blF^blF^bl-%*LINESTYLEG6#\"\"$-F$6%7$7$$\"3!*******
*******R$F-$!3/+++++++=F-7$F`_m$!31+++V@Ky&)F*Ff^mFh^m-F$6%7$7$$\"3#)*
***********\\?F-$!33+++++++;F-7$F[`m$!3/+++dk`X7F-Ff^mFh^m-F$6$7$Fc^m7
$F_^m$\"3$*******)=f-!=F--%&COLORG6&Fdbl$F]bl!\"\"F[amF[am-F$6$7$Fd_m7
$F`_m$\"31+++uwhj:F-Fh`m-%)POLYGONSG6(7&7$$F\\clF^bl$\"+g.xT=!\"*7$$\"
+&ech-#F[bm$\"+4P7@=F[bm7$F]bm$!+CT'zB\"F[bm7$Fham$!+iXWH7F[bm7&F\\bm7
$$\"+!H:*[?F[bm$\"+mR!H!=F[bm7$Fibm$!+-n>X7F[bmFabm7&Fhbm7$$\"+M)4X2#F
[bm$\"+PB@#y\"F[bm7$Fbcm$!+Aa2`7F[bmF]cm7&Facm7$$\"+++++@F[bm$\"+2%z9w
\"F[bm7$F[dm$!+W\")fg7F[bmFfcm-Fi`m6&Fdbl$\"\"'F\\amFddmFddm-%&STYLEG6
#%,PATCHNOGRIDG-Fdam6<7&7$$\"+++++q!#5$\"+*=f-!=F[bm7$$\"+Dj/x\")F`em$
\"+e+,O=F[bm7$Fdem$!+J-X)[\"F[bm7$F^em$!+z1a7;F[bm7&Fcem7$$\"+R!)=,#*F
`em$\"+O&>*y=F[bm7$F`fm$!+)Hu$*Q\"F[bmFhem7&F_fm7$$\"+^UHN5F[bm$\"+9=!
R$>F[bm7$Fifm$!+W=Z#H\"F[bmFdfm7&Fhfm7$$\"+bJB^6F[bm$\"+H@o))>F[bm7$Fb
gm$!+_uF97F[bmF]gm7&Fagm7$$\"+A5im7F[bm$\"+-n`M?F[bm7$F[hm$!+i8Ee6F[bm
Ffgm7&Fjgm7$$\"+Y.gt8F[bm$\"+H(3I1#F[bm7$Fdhm$!+5eaE6F[bmF_hm7&Fchm7$$
\"+R7P%[\"F[bm$\"+fMQt?F[bm7$F]im$!+Q[`86F[bmFhhm7&F\\im7$$\"+b1$*)f\"
F[bm$\"+sjeg?F[bm7$Ffim$!+*R+\">6F[bmFaim7&Feim7$$\"+xE78<F[bm$\"+4IQB
?F[bm7$F_jm$!+bEGS6F[bmFjim7&F^jm7$$\"+3KeI=F[bm$\"+V#4='>F[bm7$Fhjm$!
+W9Qt6F[bmFcjm7&Fgjm7$$\"+3D/M>F[bm$\"+S]$=*=F[bm7$Fa[n$!+0US27F[bmF\\
[n7&F`[n7$$\"+vJ^]?F[bm$\"+$e<;!=F[bm7$Fj[n$!+EqpX7F[bmFe[n7&Fi[n7$$\"
+$3iu;#F[bm$\"+FLn1<F[bm7$Fc\\n$!+6#R&y7F[bmF^\\n7&Fb\\n7$$\"+DS;!G#F[
bm$\"+fy,>;F[bm7$F\\]n$!+\"*o1+8F[bmFg\\n7&F[]n7$$\"+&=3DQ#F[bm$\"+iTu
[:F[bm7$Fe]n$!+e]#zI\"F[bmF`]n7&Fd]n7$$\"+!H0U]#F[bm$\"+mV\"H[\"F[bm7$
F^^n$!+=-H+8F[bmFi]n7&F]^n7$$\"+-()H2EF[bm$\"+_uhX9F[bm7$Fg^n$!+Wqfy7F
[bmFb^n7&Ff^n7$$\"+[3AFFF[bm$\"+8'=^U\"F[bm7$F`_n$!+k/ZO7F[bmF[_n7&F__
n7$$\"+nAPLGF[bm$\"+#>*\\E9F[bm7$Fi_n$!+hT)f=\"F[bmFd_n7&Fh_n7$$\"+)>P
)\\HF[bm$\"+SY[X9F[bm7$Fb`n$!+`>!*>6F[bmF]`n7&Fa`n7$$\"+a$R21$F[bm$\"+
e#=^Z\"F[bm7$F[an$!+Zp+^5F[bmFf`n7&Fj`n7$$\"+>TXwJF[bm$\"+*f\"y5:F[bm7
$Fdan$!+%o5Ry*F`emF_an7&Fcan7$$\"+&R;FG$F[bm$\"+i-*3a\"F[bm7$F]bn$!+ce
/h\"*F`emFhan7&F\\bn7$$\"+++++MF[bm$\"+twhj:F[bm7$Ffbn$!+R@Ky&)F`emFab
n-Fi`m6&Fdbl$\"#&)!\"#F_cnF_cnFfdm-F$6$7$7$Fham$!3:5zbhXWH7F-7$Fham$\"
3DLnTg.xT=F-Ff^m-F$6$7$7$$\"33+++++++@F-$!3'******H9)fg7F-7$F_dn$\"3%*
*****pSz9w\"F-Ff^m-F$6%7$7$FhblF``mF_`mFf^mFh^m-F$6%7$7$Fhbl$\"3/+++33
.-=F-7$F[`mF^enFf^mFh^m-F$6&7$7$$\"#=F\\am$\"+b<Z#y#F`em7$$\"#?F\\amFg
en7%7$$\"++++S>F[bm$\"+b<Z#G$F`emFien7$F^fn$\"+b<Z#G#F`emFfdmFf^m-F$6&
7$7$$\"#BF\\amFgen7$$\"#@F\\amFgen7%7$$\"++++g@F[bmFcfnF[gn7$F`gnF`fnF
fdmFf^m-F$6&7$7$$F\\clF\\am$\"+b<Z#G%F`em7$Fggn$\"+33.-=F[bm7%7$$\"+++
+];F`em$\"+`f'eq\"F[bmFjgn7$$\"++++]BF`emFahnFfdmFf^m-F$6&7$7$Fggn$\"+
b<Z#G\"F`em7$Fggn$!+ck`X7F[bm7%7$Fdhn$!+-;P\\6F[bmF\\in7$F_hnFainFfdmF
f^m-%%TEXTG6%7$$\"$v$FacnFeenQ)y~=~f(x)6\"Fabl-Fein6%7$Fhin$!$0\"FacnQ
)y~=~g(x)F[jnFi]m-Fein6%7$$\"\"(F\\am$!#AF\\amQ$x=aF[jnFf^m-Fein6%7$$
\"#MF\\am$!#>F\\amQ$x=bF[jnFf^m-Fein6%7$FggnFgenQ,f(x)~-~g(x)F[jnFf^m-
Fein6%7$$\"$Z#FacnFgenQ\"xF[jnFf^m-Fein6%7$$\"++++]?F[bm$F-F\\amF[\\oF
f^m-Fein6%7$$\"#CF\\amFgenQ\"DF[jn-%%FONTG6$%'SYMBOLG\"#5-%+AXESLABELS
G6%F[\\oQ\"yF[jn-Fi\\o6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;$Fa
cnF\\am$\"$.%Facn;FgjnF\\gn" 1 2 0 1 10 0 2 9 1 1 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur
ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1
8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "
Curve 25" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 40 "The narrow dark grey shaded strip is an " }{TEXT 
259 12 "area element" }{TEXT -1 33 " which represents a typical term \+
" }{XPPEDIT 18 0 "(f(x)-g(x))*`.`*Delta*x;" "6#**,&-%\"fG6#%\"xG\"\"\"
-%\"gG6#F(!\"\"F)%\".GF)%&DeltaGF)F(F)" }{TEXT -1 12 " in the sum " }
{XPPEDIT 18 0 "Sum((f(x)-g(x))*`.`*Delta*x,x = a .. b);" "6#-%$SumG6$*
*,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F+!\"\"F,%\".GF,%&DeltaGF,F+F,/F+;%\"aG%
\"bG" }{TEXT -1 303 ". Since we are considering the strip to be very n
arrow, we shall not worry about whether we \"flatten the top and botto
m\" so that we have a genuine rectangle, or just think of the top and \+
bottom as following small segments of the respective curves. Thus we t
hink of the strip as simply having the height " }{XPPEDIT 18 0 "f(x)-g
(x);" "6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F'!\"\"" }{TEXT -1 8 ", where " 
}{TEXT 288 1 "x" }{TEXT -1 27 " is its location along the " }{TEXT 
289 1 "x" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 8 "The sum \+
" }{XPPEDIT 18 0 "Sum((f(x)-g(x))*`.`*Delta*x,x = a .. b);" "6#-%$SumG
6$**,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F+!\"\"F,%\".GF,%&DeltaGF,F+F,/F+;%\"
aG%\"bG" }{TEXT -1 60 " can be pictured by \"slicing\" the region betw
een the graphs " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"gG6#%\"xG" }
{TEXT -1 6 " from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 4 "
 to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 140 " into narrow
 strips by equally spaced vertical lines as in the earlier picture. Ea
ch narrow strip is regarded as having a specific location " }{TEXT 
287 1 "x" }{TEXT -1 11 " and width " }{XPPEDIT 18 0 "Delta*x;" "6#*&%&
DeltaG\"\"\"%\"xGF%" }{TEXT -1 81 ". We can then think of each strip a
s being an approximate rectangle with area of " }{XPPEDIT 18 0 "(f(x)-
g(x))*`.`*Delta*x;" "6#**,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F(!\"\"F)%\".GF)
%&DeltaGF)F(F)" }{TEXT -1 41 " so that the total area of the region is
 " }{XPPEDIT 18 0 "Sum((f(x)-g(x))*`.`*Delta*x,x = a .. b);" "6#-%$Sum
G6$**,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F+!\"\"F,%\".GF,%&DeltaGF,F+F,/F+;%
\"aG%\"bG" }{TEXT -1 86 ". As we increase the number of strips the app
roximation successively improves so that " }{XPPEDIT 18 0 "Sum((f(x)-g
(x))*`.`*Delta*x,x = a .. b);" "6#-%$SumG6$**,&-%\"fG6#%\"xG\"\"\"-%\"
gG6#F+!\"\"F,%\".GF,%&DeltaGF,F+F,/F+;%\"aG%\"bG" }{TEXT -1 12 " appro
aches " }{XPPEDIT 18 0 "Int(``(f(x)-g(x)),x = a .. b);" "6#-%$IntG6$-%
!G6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F-!\"\"/F-;%\"aG%\"bG" }{TEXT -1 75 "
 as the number of strips tends to infinity, and their width tends to z
ero. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 46 "The area between two graphs . . more exam
ples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT 292 8 "Ques
tion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 65 "Calculate the ar
ea of the region enclosed between the graphs of  " }{XPPEDIT 18 0 "y =
 x^3+x;" "6#/%\"yG,&*$%\"xG\"\"$\"\"\"F'F)" }{TEXT -1 7 "  and  " }
{XPPEDIT 18 0 "y = x^2+x+1;" "6#/%\"yG,(*$%\"xG\"\"#\"\"\"F'F)F)F)" }
{TEXT -1 6 " from " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"\"!\"\"" }
{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 8 "S
olution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 43 "This region c
an be illustrated as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 621 "f := x -> x^2+x+1:\ng := x \+
-> x^3+x:\na := -1: b := 1:\nc := -1.4: d := 1.4:\np1 := plot([f(x),g(
x)],x=c..d,color=[red,blue],thickness=2):\np2 := plot([[[a,g(a)],[a,f(
a)]],[[b,g(b)],[b,f(b)]]],\n                color=COLOR(RGB,.4,.4,.4))
:\np3 := plot([[b,0],[b,g(b)]],color=COLOR(RGB,.5,.5,.5),linestyle=3):
\npp := plot(f(x),x=a..b,adaptive=false,numpoints=25):\nfpts := op(1,o
p(1,pp)):\npp := plot(g(x),x=a..b,adaptive=false,numpoints=25):\ngpts \+
:= op(1,op(1,pp)):\np4 := plots[polygonplot]([seq([fpts[i-1],fpts[i],g
pts[i],gpts[i-1]],i=2..25)],\n           style=patchnogrid,color=COLOR
(RGB,.9,.9,.9)):\nplots[display]([p1,p2,p3,p4]);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 335 446 446 {PLOTDATA 2 "6*-%'CURVESG6%7S7$$!3!*************
*R\"!#<$\"3%)************f:F*7$$!3JLLLoz'*Q8F*$\"3%4O*)=DnQX\"F*7$$!3]
mm\"RKkeG\"F*$\"3?**\\gNFen8F*7$$!3JLL$=sVhA\"F*$\"3Slzv/ZGx7F*7$$!3IL
L$GDFg;\"F*$\"3!pr;:I#f$>\"F*7$$!3gmm\"p]'>16F*$\"3Coz*\\?uu6\"F*7$$!3
GLLeYds]5F*$\"3!)p(\\z%))H`5F*7$$!3A****\\s)*)G$**!#=$\"3M18U\\-ML**FN
7$$!37LL$e3y)Q$*FN$\"3KtCP/je#Q*FN7$$!3M****\\-8xY()FN$\"3x`PK>&HQ!*)F
N7$$!3'emmmp;x8)FN$\"3+xS&ogEX[)FN7$$!3fKLLo5E,wFN$\"3;nEc9flw\")FN7$$
!3')*******3Rt*pFN$\"3%G?WSMO*)*yFN7$$!3'*******\\t$4R'FN$\"3E-D;r1Z$p
(FN7$$!3=******4pb1eFN$\"31uo!\\S`]c(FN7$$!3mLL$e[$)eF&FN$\"3Kabxp6h2v
FN7$$!3$ommmXh[k%FN$\"3*=4'\\QBh7vFN7$$!3DmmmEII5TFN$\"3g\\eVqg:zvFN7$
$!3O,+]#\\%[)[$FN$\"34)HTH\"zYGxFN7$$!3xnmmE(p!QHFN$\"3vY)3_kb^#zFN7$$
!3G++]#\\xTL#FN$\"3()e$\\T'4m5#)FN7$$!3/++]x#H\"f<FN$\"3gmd*RIC.b)FN7$
$!3+nmm\"zE\"f6FN$\"3Ez#\\-qI_(*)FN7$$!3;fmm\"4s83'!#>$\"3iN1Nxe%)G%*F
N7$$!3#[JLL$30#Q\"!#?$\"3!es6Bf)>')**FN7$$\"3/dmmT^2NgFir$\"3!HA8Y'H*R
1\"F*7$$\"37)***\\sL*39\"FN$\"32WT79d5F6F*7$$\"3_mmm@[G@<FN$\"35n'Rlpc
<?\"F*7$$\"38)*****zf)3K#FN$\"3S0;K:P&fG\"F*7$$\"3a(****\\tyu!HFN$\"3q
rZW*>#Gv8F*7$$\"3e****\\(QP]Z$FN$\"3=G_>(ei#o9F*7$$\"3&))****\\]9_5%FN
$\"3qB1#=J\\!z:F*7$$\"3eHLLtmXrYFN$\"3#zjT=uq`o\"F*7$$\"3!*******\\5/w
_FN$\"3'4&Gh'>qf!=F*7$$\"31km;aV)Q#eFN$\"35RP7DZc@>F*7$$\"3`*******y@G
U'FN$\"3!y)4Yw&3[0#F*7$$\"31HL$3Cvj)pFN$\"3sA199>t'=#F*7$$\"33++]x4Xvv
FN$\"3\"p0vGn>9L#F*7$$\"39KLL$=!Q^\")FN$\"3#osls+)ezCF*7$$\"3#p***\\(o
CVv)FN$\"3/><4wW\"=k#F*7$$\"3romm1>.N$*FN$\"3A9:l(RJ\\!GF*7$$\"3akmm6*
)))G**FN$\"3wQ)o8C<(yHF*7$$\"37LL3[Gy^5F*$\"39@cg2+.eJF*7$$\"3,+++y-!f
5\"F*$\"3Ax/)Gq:*GLF*7$$\"3WLL$)f\\#z;\"F*$\"3'p-N;nt>`$F*7$$\"3_mmmu0
SB7F*$\"3W'Rh2a4,s$F*7$$\"3)****\\2:\\DG\"F*$\"38Y&R[Z\"[FRF*7$$\"3z**
*\\_)=;R8F*$\"3YLj:Sk^KTF*7$$\"3!**************R\"F*$\"3U************f
VF*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fb[lFa[l-%*THICKNESSG6#\"\"#-
F$6%7S7$F($!3C***********R9%F*7$F.$!3Xe8r+p^RPF*7$F3$!3d$\\S#4H(>T$F*7
$F8$!3#3-C$[IcpIF*7$F=$!33.1p12Q^FF*7$FB$!3CmG0un\")fCF*7$FG$!3')*[?H
\"Gv5AF*7$FL$!3]ly#)f2Ht>F*7$FR$!35@SGNZP[<F*7$FW$!304@S(odQa\"F*7$Ffn
$!3Ek6no5n_8F*7$F[o$!3!o#[,h1K*>\"F*7$F`o$!3:U/!R!HMU5F*7$Feo$!3)Gq3v#
pD,!*FN7$Fjo$!3Sx\\=F;IkxFN7$F_p$!3@ilZMGUWnFN7$Fdp$!3'>leT9ypk&FN7$Fi
p$!3lm;p5>s/[FN7$F^q$!32c#\\>X;I\"RFN7$Fcq$!3cqk!*f8p\">$FN7$Fhq$!3SPf
veCNhCFN7$F]r$!3aVO-jhc8=FN7$Fbr$!3/e2_G0qu6FN7$Fgr$!3kgwo#)G'Q5'Fir7$
F]s$!3!pbRJZ`?Q\"F_s7$Fcs$\"3;W)*)><cq0'Fir7$Fhs$\"3Yr'41nVd:\"FN7$F]t
$\"3G)**ftT$Gs<FN7$Fbt$\"3KnFgv2!fW#FN7$Fgt$\"3Q!)>1!\\gK:$FN7$F\\u$\"
3!Rh=2!*yY*QFN7$Fau$\"3=_+zDv0(z%FN7$Ffu$\"3u8Q$Rl&)3p&FN7$F[v$\"36iuF
w>rWnFN7$F`v$\"3=%)f4Ty?*z(FN7$Fev$\"3]LEcC`Ss!*FN7$Fjv$\"38A>F!eQ'R5F
*7$F_w$\"3=^$f)o4G#>\"F*7$Fdw$\"3a*G$flkvc8F*7$Fiw$\"3YTFWh\"[ja\"F*7$
F^x$\"3eaRK-U)pu\"F*7$Fcx$\"3cl&z'3pqr>F*7$Fhx$\"3(>y4ij9`@#F*7$F]y$\"
393OPwLVeCF*7$Fby$\"3CO]x'\\J5w#F*7$Fgy$\"3;6d*Q=vW0$F*7$F\\z$\"3KrA01
dD#R$F*7$Faz$\"3!*R+#>&RvSPF*7$Ffz$\"3C***********R9%F*-F[[l6&F][lFa[l
Fa[lF^[lFc[l-F$6$7$7$$!\"\"Fb[l$!\"#Fb[l7$Fcel$\"\"\"Fb[l-%&COLORG6&F]
[l$\"\"%FdelF]flF]fl-F$6$7$7$Fhel$Ff[lFb[l7$Fhel$\"\"$Fb[lFjel-F$6%7$7
$FhelFa[lFbfl-F[fl6&F][l$\"\"&FdelF]glF]gl-%*LINESTYLEG6#Fffl-%)POLYGO
NSG6<7&Fgel7$$!+LQ6G\"*!#5$\"+#QKT?*Figl7$Fggl$!+.$)Qt;!\"*Fbel7&Ffgl7
$$!+U.\\p$)Figl$\"+;lMN')Figl7$Fbhl$!+y\"=KU\"F_hlF\\hl7&Fahl7$$!+$))Q
j^(Figl$\"+Qh>L\")Figl7$F[il$!+'3si<\"F_hlFfhl7&Fjhl7$$!+$=Kvl'Figl$\"
+%HTZx(Figl7$Fdil$!+:DL3'*FiglF_il7&Fcil7$$!+Us!G!eFigl$\"+Z*\\Wc(Figl
7$F]jl$!+\"ohnv(FiglFhil7&F\\jl7$$!+3yO5]Figl$\"+\\2,+vFigl7$Ffjl$!+*z
f\"oiFiglFajl7&Fejl7$$!+vE%)*=%Figl$\"+*[Ncc(Figl7$F_[m$!+3/OD\\FiglFj
jl7&F^[m7$$!+3WDTLFigl$\"+%pV^x(Figl7$Fh[m$!+;9F9PFiglFc[m7&Fg[m7$$!+v
vQ&\\#Figl$\"+S$3t7)Figl7$Fa\\m$!+SVx]EFiglF\\\\m7&F`\\m7$$!+n&4`i\"Fi
gl$\"+_N&)Q')Figl7$Fj\\m$!+^UCo;FiglFe\\m7&Fi\\m7$$!+LQW*e)!#6$\"+iT$[
@*Figl7$Fc]m$!++b\"Gl)Fe]mF^]m7&Fb]m7$$\"+++I,Q!#8$\"+u-Q+5F_hl7$F]^m$
\"+\\0I,QF_^mFh]m7&F\\^m7$$\"++]*3q)Fe]m$\"+2&zX4\"F_hl7$Fg^m$\"+Dcwm(
)Fe]mFb^m7&Ff^m7$$\"++(=\\q\"Figl$\"+[$f&*>\"F_hl7$F`_m$\"+'QwWv\"Figl
F[_m7&F__m7$$\"+#fBIY#Figl$\"+6s'pI\"F_hl7$Fi_m$\"+aDW7EFiglFd_m7&Fh_m
7$$\"+LO[kLFigl$\"+lek\\9F_hl7$Fb`m$\"+tZLXPFiglF]`m7&Fa`m7$$\"+L&Q\"G
TFigl$\"+J\"HKe\"F_hl7$F[am$\"+2kjJ[FiglFf`m7&Fj`m7$$\"+D2X;]Figl$\"+^
GH`<F_hl7$Fdam$\"+2%H)yiFiglF_am7&Fcam7$$\"+Lvv-eFigl$\"+.d*p\">F_hl7$
F]bm$\"+f<mcxFiglFham7&F\\bm7$$\"+D2YlmFigl$\"+R(H36#F_hl7$Ffbm$\"+dg
\"oi*FiglFabm7&Febm7$$\"+v\"ep[(Figl$\"+X7C4BF_hl7$F_cm$\"+MQPo6F_hlFj
bm7&F^cm7$$\"+$e/TM)Figl$\"+r7lIDF_hl7$Fhcm$\"+u5O:9F_hlFccm7&Fgcm7$$
\"+eDBJ\"*Figl$\"+Ot\"pu#F_hl7$Fadm$\"+@+[u;F_hlF\\dm7&F`dmFdflFbflFed
m-F[fl6&F][l$\"\"*FdelF[emF[em-%&STYLEG6#%,PATCHNOGRIDG-%+AXESLABELSG6
%Q\"x6\"Q!Feem-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!#9Fdel$\"#9FdelFjem" 1 
2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve
 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 14 "The graph of  " }{XPPEDIT 18 0 "g(x) = x^3+x;" "6#/-%
\"gG6#%\"xG,&*$F'\"\"$\"\"\"F'F+" }{TEXT -1 13 " is drawn in " }{TEXT 
256 4 "blue" }{TEXT -1 22 ", while the graph of  " }{XPPEDIT 18 0 "f(x
) = x^2+x+1;" "6#/-%\"fG6#%\"xG,(*$F'\"\"#\"\"\"F'F+F+F+" }{TEXT -1 
13 " is drawn in " }{TEXT 260 3 "red" }{TEXT -1 2 ". " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The picture shows that
 " }{XPPEDIT 18 0 "g(x)<f(x)" "6#2-%\"gG6#%\"xG-%\"fG6#F'" }{TEXT -1 
5 " for " }{XPPEDIT 18 0 "-1<=x" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 
0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 61 ", so the area of the region enc
losed between the curves from " }{XPPEDIT 18 0 "x=-1" "6#/%\"xG,$\"\"
\"!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }
{TEXT -1 14 " is given by  " }{XPPEDIT 18 0 "Int(``(f(x)-g(x)),x = -1 \+
.. 1);" "6#-%$IntG6$-%!G6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F-!\"\"/F-;,$F.
F2F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 5 "Now  " }{XPPEDIT 18 0 "f(x)-g(x) = 1+x^2-x^3;" "6#/,&-%
\"fG6#%\"xG\"\"\"-%\"gG6#F(!\"\",(F)F)*$F(\"\"#F)*$F(\"\"$F-" }{TEXT 
-1 27 ", so the required area is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "Int(``(1+x^2-x^3),x = -1 .. 1) = x+x^3/3-x^4/4;" "6#
/-%$IntG6$-%!G6#,(\"\"\"F+*$%\"xG\"\"#F+*$F-\"\"$!\"\"/F-;,$F+F1F+,(F-
F+*&F-F0F0F1F+*&F-\"\"%F8F1F1" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "PIECEW
ISE([1, ``],[-1, ``]);" "6#-%*PIECEWISEG6$7$\"\"\"%!G7$,$F'!\"\"F(" }
{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` =
 ``(1+1/3-1/4)-(-1-1/3-1/4);" "6#/%!G,&-F$6#,(\"\"\"F)*&F)F)\"\"$!\"\"
F)*&F)F)\"\"%F,F,F),(F)F,*&F)F)F+F,F,*&F)F)F.F,F,F," }{TEXT -1 1 " " }
}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2+2/3;" "6#/%!G
,&\"\"#\"\"\"*&F&F'\"\"$!\"\"F'" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2;" "6#/%!G\"\"#" }{TEXT -1 1 " " 
}{XPPEDIT 18 0 "2/3 = 8/3;" "6#/*&\"\"#\"\"\"\"\"$!\"\"*&\"\")F&F'F(" 
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 95 "f := x -> x^2+x+1: 'f(x)'=f(x);\ng := x -> x^3
+x: 'g(x)'=g(x);\nInt(f(x)-g(x),x=-1..1);\nvalue(%);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*$)F'\"\"#\"\"\"F,F'F,F,F," }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&*$)F'\"\"$\"\"\"F,F'F," }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,(*$)%\"xG\"\"#\"\"\"F+F+F+*$
)F)\"\"$F+!\"\"/F);F/F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\")\"\"$
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 \+
" }}{PARA 0 "" 0 "" {TEXT 294 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 
"" 0 "" {TEXT -1 59 "Calculate the area of the region bounded by the g
raphs of  " }{XPPEDIT 18 0 "y = x^2-2*x-3;" "6#/%\"yG,(*$%\"xG\"\"#\"
\"\"*&F(F)F'F)!\"\"\"\"$F+" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "y = \+
6+x-x^2;" "6#/%\"yG,(\"\"'\"\"\"%\"xGF'*$F(\"\"#!\"\"" }{TEXT -1 38 " \+
between their points of intersection." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 295 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 
"" 0 "" {TEXT -1 29 "The two parabolas meet where " }{XPPEDIT 18 0 "x^
2-2*x-3 = 6+x-x^2;" "6#/,(*$%\"xG\"\"#\"\"\"*&F'F(F&F(!\"\"\"\"$F*,(\"
\"'F(F&F(*$F&F'F*" }{TEXT -1 17 ", that is, where " }{XPPEDIT 18 0 "2*
x^2-3*x-9 = 0;" "6#/,(*&\"\"#\"\"\"*$%\"xGF&F'F'*&\"\"$F'F)F'!\"\"\"\"
*F,\"\"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "(2*x+3)*(x-3) = 0;" "6#/*
&,&*&\"\"#\"\"\"%\"xGF(F(\"\"$F(F(,&F)F(F*!\"\"F(\"\"!" }{TEXT -1 2 ".
 " }}{PARA 0 "" 0 "" {TEXT -1 38 "This shows that the graphs meet wher
e " }{XPPEDIT 18 0 "x = -3/2;" "6#/%\"xG,$*&\"\"$\"\"\"\"\"#!\"\"F*" }
{TEXT -1 11 " and where " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 103 "f := x -> 6+x-x^2: 'f(x)'=f(x);\ng := x -> x^2-2*x-3
: 'g(x)'=g(x);\n'f(x)-g(x)'=f(x)-g(x);\nsolve(rhs(%));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(\"\"'\"\"\"F'F**$)F'\"\"#F*!\"\"" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,(*$)F'\"\"#\"\"\"F,*&
F+F,F'F,!\"\"\"\"$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%\"fG6#%\"
xG\"\"\"-%\"gGF'!\"\",(\"\"*F)*&\"\"$F)F(F)F)*&\"\"#F))F(F2F)F," }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$#!\"$\"\"#\"\"$" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The region can be illustr
ated as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 524 "f := x -> 6+x-x^2:\ng := x -> x^2-2*x-3:\na \+
:= -1.5: b := 3:\nc := -2: d := 3.5:\np1 := plot([f(x),g(x)],x=c..d,co
lor=[red,blue],thickness=2):\np2 := plot([[a,0],[a,g(a)]],color=COLOR(
RGB,.5,.5,.5),linestyle=3):\npp := plot(f(x),x=a..b,adaptive=false,num
points=25):\nfpts := op(1,op(1,pp)):\npp := plot(g(x),x=a..b,adaptive=
false,numpoints=25):\ngpts := op(1,op(1,pp)):\np3 := plots[polygonplot
]([seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2..25)],\n           st
yle=patchnogrid,color=COLOR(RGB,.9,.9,.9)):\nplots[display]([p1,p2,p3]
);" }}{PARA 13 "" 1 "" {GLPLOT2D 341 393 393 {PLOTDATA 2 "6(-%'CURVESG
6%7S7$$!\"#\"\"!$F*F*7$$!3ILL3_c6!)=!#<$\"3O$Hzq#[\\]e!#=7$$!3um\"z>#
\\!ex\"F/$\"3o)R&4d>rq5F/7$$!3OL$ekf'\\e;F/$\"33\"zxIW#*3f\"F/7$$!3IL$
3_n5/a\"F/$\"3cte\\wUs'3#F/7$$!3jm\"Hd*f)GU\"F/$\"3;^'GuW4Db#F/7$$!3HL
ektb#RJ\"F/$\"3KMOF8SnfHF/7$$!3/+D\"yO.6?\"F/$\"37qp,KtCcLF/7$$!3SLe9
\"[AW3\"F/$\"3b\"oR7S0'RPF/7$$!3]+]i:z:\"o*F2$\"3]8[cAgj%4%F/7$$!3]mm;
a1![[)F2$\"33jQ<8:gJWF/7$$!3kKL3F&[5V(F2$\"3o$*H<D.p/ZF/7$$!3%))***\\7
KxWiF2$\"3/DW*Rv]b)\\F/7$$!3--+]Ppi`]F2$\"3%Hu:S&eCR_F/7$$!3U++](yOd!R
F2$\"3#>p<F_yoX&F/7$$!3AN$e9cUL'GF2$\"3%*y:hP%y;j&F/7$$!3:nm;/]$Qi\"F2
$\"3]jdP)4[7\"eF/7$$!32zmmm^4Qd!#>$\"3?h\">ZZE$RfF/7$$\"3/*)*\\(ou>wkF
[q$\"3#[2A8'ycggF/7$$\"3zGLL3;zG<F2$\"3*\\*Gec>*H9'F/7$$\"33+](o\\3]\"
HF2$\"3^l*=V5Gl?'F/7$$\"3w,]i!\\nX/%F2$\"3h%H+s[r3C'F/7$$\"3HML3-Q9B_F
2$\"3%z\"fVo?]\\iF/7$$\"3tM$3xwWaI'F2$\"3>HYeR\"eHB'F/7$$\"3%Rm;Hd_GZ(
F2$\"3o%>E:+]))='F/7$$\"3Ej;a)=hao)F2$\"3iIvFeP<9hF/7$$\"3%***\\7`0/T(
*F2$\"30UitW`ADgF/7$$\"3qm;a=&4\")3\"F/$\"3@*o+U>FT!fF/7$$\"3Z****\\<$
))e?\"F/$\"3s>q@$o@<v&F/7$$\"3a**\\P%*=6@8F/$\"3O**e\"pDvdd&F/7$$\"3o*
\\(oHmfK9F/$\"3u;bHENE!Q&F/7$$\"3u**\\i?9Qc:F/$\"3&zIeTHeS8&F/7$$\"3%H
LL3c2wm\"F/$\"3uM[)Qe#p')[F/7$$\"3s***\\i?ljy\"F/$\"35'R\\hbk_f%F/7$$
\"3SmT&QGxR*=F/$\"3Rg&)4Kt#oI%F/7$$\"3Y++v3di6?F/$\"3)y0Ymr()\\'RF/7$$
\"3\"G$3-3PKA@F/$\"3rNNR'yl!=OF/7$$\"3++vV8].QAF/$\"3w3`1#HM#HKF/7$$\"
3)GL$ekR;^BF/$\"3d;\"Gi(>>BGF/7$$\"3I*\\(=#\\*fpCF/$\"3sVZPSynqBF/7$$
\"33nmm\")pm$e#F/$\"3<@@^4>L3>F/7$$\"3dm;zkuJ+FF/$\"3?tF8aIg39F/7$$\"3
OL3-B?+;GF/$\"3MgSj$HY8'))F27$$\"3T++]<TIAHF/$\"31daUBwUCQF27$$\"3ML$e
RJQT/$F/$!35/v]0wREAF27$$\"3_mmmr#3J:$F/$!3AY3'=,N)*)yF27$$\"3o*\\(ouH
HpKF/$!3%>a`'zN)*=9F/7$$\"3M+DJgl\\!Q$F/$!3'\\9sRVgs/#F/7$$\"3++++++++
NF/$!3+++++++]FF/-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-%*THICKNESSG6#\"
\"#-F$6%7S7$F($\"\"&F*7$F-$\"3k/aPph1&H%F/7$F4$\"30oP)['H40PF/7$F9$\"3
;V0Q`TgnIF/7$F>$\"3ufCr)R'o`CF/7$FC$\"3[:0I[lPq=F/7$FH$\"3'*)>s.c^UN\"
F/7$FM$\"3Y,`&zNg&[%)F27$FR$\"3Y=:1*zq\"[MF27$FW$!35RJ-5B?l7F27$Ffn$!3
))f>dxW@JeF27$F[o$!3_(fYYsaeh*F27$F`o$!3sCWuKM2h8F/7$Feo$!3>VdEgJ)Qt\"
F/7$Fjo$!3)=pnR%[Im?F/7$F_p$!3UXdY\"=W`M#F/7$Fdp$!3-(4fzfk)[EF/7$Fip$!
3=%\\_I_X>)GF/7$F_q$!3fv&4g$)H`7$F/7$Fdq$!3JGiT<6(eJ$F/7$Fiq$!3_lk+a*G
!)\\$F/7$F^r$!3M%ziiBG`k$F/7$Fcr$!3O^Uk[k\"=x$F/7$Fhr$!3miaN;E]jQF/7$F
]s$!3kgy\")e_8ORF/7$Fbs$!3'ppJr()>F)RF/7$Fgs$!3/U([+SH$**RF/7$F\\t$!3#
fNUFrOA*RF/7$Fat$!3=>qr++hdRF/7$Fft$!3M**3H^r)o*QF/7$F[u$!3U;I)f:gG\"Q
F/7$F`u$!3A2Ly9(R/p$F/7$Feu$!3on\"=Z9+Vb$F/7$Fju$!3Q&R*Ri(H;Q$F/7$F_v$
!3DFF&fh/3?$F/7$Fdv$!3AfgRDMhwHF/7$Fiv$!3_oVT%\\*QSFF/7$F^w$!3w3G]0$ps
Y#F/7$Fcw$!3M]9\"3%fNu@F/7$Fhw$!3!RCiDLx-%=F/7$F]x$!3E)yy6*))*>\\\"F/7
$Fbx$!3xRW#*=0#*36F/7$Fgx$!3z-C%Q_m8-(F27$F\\y$!3E_aU)zou/$F27$Fay$\"3
oqT#fY9]y\"F27$Ffy$\"3%**=%>&H_(ejF27$F[z$\"3OTg'\\g!p\\6F/7$F`z$\"3]X
'fO(Qwm;F/7$Fez$\"3+++++++]AF/-Fjz6&F\\[lF+F+F][lF`[l-F$6%7$7$$!3+++++
+++:F/F+7$F`elFhdl-%&COLORG6&F\\[l$Fi[l!\"\"FfelFfel-%*LINESTYLEG6#\"
\"$-%)POLYGONSG6<7&7$$!+++++:!\"*$\"++++]AFcfl7$$!+8c#QI\"Fcfl$\"+gJ@'
*HFcfl7$Fgfl$\"+`Ch28FcflF`fl7&Fffl7$$!+F`8L6Fcfl$\"+/!pGe$Fcfl7$F`gl$
\"+GKm-b!#5F[gl7&F_gl7$$!+([i<T*Fggl$\"+?5,tTFcfl7$Fjgl$!+8xM=BFgglFdg
l7&Figl7$$!+7uWzuFggl$\"+BRj#p%Fcfl7$Fchl$!+:=*oW*FgglF^hl7&Fbhl7$$!+%
H;jb&Fggl$\"+j=kN^Fcfl7$F\\il$!+M-,!e\"FcflFghl7&F[il7$$!+pvKtPFggl$\"
+MsG![&Fcfl7$Feil$!+xW&H5#FcflF`il7&Fdil7$$!+>g9F>Fggl$\"+![Y,x&Fcfl7$
F^jl$!+y=VxDFcflFiil7&F]jl7$$!+v=C#y\"!#7$\"+#e9#)*fFcfl7$Fgjl$!+SBV'*
HFcflFbjl7&Ffjl7$$\"+czP&)=Fggl$\"+&H\"*H:'Fcfl7$Fa[m$!+\"4H:M$FcflF\\
[m7&F`[m7$$\"+vM0VQFggl$\"+ZZhOiFcfl7$Fj[m$!+&4?4i$FcflFe[m7&Fi[m7$$\"
+P^PnbFggl$\"+b3yYiFcfl7$Fc\\m$!+o$=N!QFcflF^\\m7&Fb\\m7$$\"+DHb3vFggl
$\"+A;2(='Fcfl7$F\\]m$!+:p#z$RFcflFg\\m7&F[]m7$$\"+v8qd%*Fggl$\"+&)*)G
^gFcfl7$Fe]m$!+A\"fq*RFcflF`]m7&Fd]m7$$\"+3ngL6Fcfl$\"+SDa[eFcfl7$F^^m
$!+[#\\@)RFcflFi]m7&F]^m7$$\"+3.=/8Fcfl$\"+KSH.cFcfl7$Fg^m$!+SVZ2RFcfl
Fb^m7&Ff^m7$$\"+=)3q]\"Fcfl$\"+TK$fB&Fcfl7$F`_m$!+f?%Hu$FcflF[_m7&F__m
7$$\"+q6$)y;Fcfl$\"+tqNg[Fcfl7$Fi_m$!+V#)=RNFcflFd_m7&Fh_m7$$\"+89qy=F
cfl$\"+8C=\\VFcfl7$Fb`m$!+FQ)yA$FcflF]`m7&Fa`m7$$\"+X/ib?Fcfl$\"+J]/IQ
Fcfl7$F[am$!+wam&)GFcflFf`m7&Fj`m7$$\"+j'G(\\AFcfl$\"+0'\\%)=$Fcfl7$Fd
am$!+p#y\"QCFcflF_am7&Fcam7$$\"+*elXV#Fcfl$\"+!)fX2DFcfl7$F]bm$!+q:-U>
FcflFham7&F\\bm7$$\"+JNUFEFcfl$\"+>\"pSs\"Fcfl7$Ffbm$!+]E\\^8FcflFabm7
&Febm7$$\"+Et_/GFcfl$\"+a!Q:R*Fggl7$F_cm$!+68\"oV(FgglFjbm7&F^cm7$$F[f
lF*F+FgcmFccm-Fdel6&F\\[l$\"\"*FgelF[dmF[dm-%&STYLEG6#%,PATCHNOGRIDG-%
+AXESLABELSG6%Q\"x6\"Q!Fedm-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"#NFgelF
jdm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" "Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
14 "The graph of  " }{XPPEDIT 18 0 "g(x) = x^2-2*x-3;" "6#/-%\"gG6#%\"
xG,(*$F'\"\"#\"\"\"*&F*F+F'F+!\"\"\"\"$F-" }{TEXT -1 13 " is drawn in \+
" }{TEXT 256 4 "blue" }{TEXT -1 22 ", while the graph of  " }{XPPEDIT 
18 0 "f(x) = 6+x-x^2;" "6#/-%\"fG6#%\"xG,(\"\"'\"\"\"F'F**$F'\"\"#!\"
\"" }{TEXT -1 13 " is drawn in " }{TEXT 260 3 "red" }{TEXT -1 2 ". " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The pict
ure shows that " }{XPPEDIT 18 0 "g(x) <= f(x);" "6#1-%\"gG6#%\"xG-%\"f
G6#F'" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "-3/2 <= x;" "6#1,$*&\"\"$\"
\"\"\"\"#!\"\"F)%\"xG" }{XPPEDIT 18 0 "`` <= 3;" "6#1%!G\"\"$" }{TEXT 
-1 61 ", so the area of the region enclosed between the curves from " 
}{XPPEDIT 18 0 "x = -3/2;" "6#/%\"xG,$*&\"\"$\"\"\"\"\"#!\"\"F*" }
{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 
14 " is given by  " }{XPPEDIT 18 0 "Int(``(f(x)-g(x)),x = -3/2 .. 3);
" "6#-%$IntG6$-%!G6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F-!\"\"/F-;,$*&\"\"$F
.\"\"#F2F2F7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 5 "Now  " }{XPPEDIT 18 0 "f(x)-g(x) = 9+3*x-2
*x^2;" "6#/,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F(!\"\",(\"\"*F)*&\"\"$F)F(F)F
)*&\"\"#F)*$F(F3F)F-" }{TEXT -1 27 ", so the required area is: " }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(9+3*x-2*x^2),x
 = -3/2 .. 3) = 9*x+3*x^2/2-2*x^3/3;" "6#/-%$IntG6$-%!G6#,(\"\"*\"\"\"
*&\"\"$F,%\"xGF,F,*&\"\"#F,*$F/F1F,!\"\"/F/;,$*&F.F,F1F3F3F.,(*&F+F,F/
F,F,*(F.F,*$F/F1F,F1F3F,*(F1F,*$F/F.F,F.F3F3" }{TEXT -1 2 "  " }
{XPPEDIT 18 0 "PIECEWISE([3, ``],[-3/2, ``]);" "6#-%*PIECEWISEG6$7$\"
\"$%!G7$,$*&F'\"\"\"\"\"#!\"\"F.F(" }{TEXT -1 1 " " }}{PARA 257 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(27+27/2-18)-(-27/2+``(3/2)*``
(9/4)-``(2/3)*(-27/8));" "6#/%!G,&-F$6#,(\"#F\"\"\"*&F)F*\"\"#!\"\"F*
\"#=F-F*,(*&F)F*F,F-F-*&-F$6#*&\"\"$F*F,F-F*-F$6#*&\"\"*F*\"\"%F-F*F**
&-F$6#*&F,F*F5F-F*,$*&F)F*\"\")F-F-F*F-F-" }{TEXT -1 1 " " }}{PARA 
257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(9+27/2)-(-27/2+27/8+9
/4)" "6#/%!G,&-F$6#,&\"\"*\"\"\"*&\"#FF*\"\"#!\"\"F*F*,(*&F,F*F-F.F.*&
F,F*\"\")F.F**&F)F*\"\"%F.F*F." }{TEXT -1 1 " " }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 9+27-27/8-9/4;" "6#/%!G,*\"\"*\"\"
\"\"#FF'*&F(F'\"\")!\"\"F+*&F&F'\"\"%F+F+" }{TEXT -1 1 " " }}{PARA 
257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=36-3" "6#/%!G,&\"#O\"\"
\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "3/8 - 2" "6#,&*&\"\"$\"
\"\"\"\")!\"\"F&\"\"#F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/4" "6#*&\"
\"\"F$\"\"%!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``=31 - 5/8" "6#/%!G,&\"#J\"\"\"*&\"\"&F'\"\")!\"\"F+" 
}{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=
30" "6#/%!G\"#I" }{TEXT -1 1 " " }{XPPEDIT 18 0 "3/8 = 243/8" "6#/*&\"
\"$\"\"\"\"\")!\"\"*&\"$V#F&F'F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "f := x -> 6
+x-x^2: 'f(x)'=f(x);\ng := x -> x^2-2*x-3: 'g(x)'=g(x);\nInt(f(x)-g(x)
,x=-3/2..3);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%
\"xG,(\"\"'\"\"\"F'F**$)F'\"\"#F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"gG6#%\"xG,(*$)F'\"\"#\"\"\"F,*&F+F,F'F,!\"\"\"\"$F." }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,(\"\"*\"\"\"*&\"\"$F(%\"xGF(
F(*&\"\"#F()F+F-F(!\"\"/F+;#!\"$F-F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6##\"$V#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Examp
le 3 " }}{PARA 0 "" 0 "" {TEXT 302 8 "Question" }{TEXT -1 2 ": " }}
{PARA 0 "" 0 "" {TEXT -1 70 "Calculate the total area of the region(s)
 enclosed between the curves " }{XPPEDIT 18 0 "y = 8*x-x^2;" "6#/%\"yG
,&*&\"\")\"\"\"%\"xGF(F(*$F)\"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "y = x^3-3*x^2;" "6#/%\"yG,&*$%\"xG\"\"$\"\"\"*&F(F)*$F'\"\"#F)!
\"\"" }{TEXT -1 55 " between any points of intersection of the two cur
ves. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 303 8 "
Solution" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 26 "The two curve
s meet where " }{XPPEDIT 18 0 "8*x-x^2 = x^3-3*x^2;" "6#/,&*&\"\")\"\"
\"%\"xGF'F'*$F(\"\"#!\"\",&*$F(\"\"$F'*&F.F'*$F(F*F'F+" }{TEXT -1 17 "
, that is, where " }{XPPEDIT 18 0 "x^3-2*x^2-8*x = 0;" "6#/,(*$%\"xG\"
\"$\"\"\"*&\"\"#F(*$F&F*F(!\"\"*&\"\")F(F&F(F,\"\"!" }{TEXT -1 4 " or \+
" }{XPPEDIT 18 0 "x*(x+2)*(x-4) = 0;" "6#/*(%\"xG\"\"\",&F%F&\"\"#F&F&
,&F%F&\"\"%!\"\"F&\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
38 "This shows that the graphs meet where " }{XPPEDIT 18 0 "x=-2,0" "6
$/%\"xG,$\"\"#!\"\"\"\"!" }{TEXT -1 11 " and where " }{XPPEDIT 18 0 "x
 = 4;" "6#/%\"xG\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "f := x -> 8*x-x^2: 'f(x
)'=f(x);\ng := x -> x^3-3*x^2: 'g(x)'=g(x);\n'f(x)-g(x)'=f(x)-g(x);\ns
olve(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*&\"
\")\"\"\"F'F+F+*$)F'\"\"#F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-
%\"gG6#%\"xG,&*$)F'\"\"$\"\"\"F,*&F+F,)F'\"\"#F,!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/,&-%\"fG6#%\"xG\"\"\"-%\"gGF'!\"\",(*&\"\")F)F(F)
F)*&\"\"#F))F(F1F)F)*$)F(\"\"$F)F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6%
\"\"!\"\"%!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 42 "The region can be illustrated as follows. " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 560 "f := x
 -> 8*x-x^2:\ng := x -> x^3-3*x^2:\na := -2: b := 4:\nc := -2.2: d := \+
4.2:\np1 := plot([f(x),g(x)],x=c..d,color=[red,blue],thickness=2):\np2
 := plot([[[a,0],[a,g(a)]],[[b,0],[b,g(b)]]],\n                color=C
OLOR(RGB,.5,.5,.5),linestyle=3):\npp := plot(f(x),x=a..b,adaptive=fals
e,numpoints=25):\nfpts := op(1,op(1,pp)):\npp := plot(g(x),x=a..b,adap
tive=false,numpoints=25):\ngpts := op(1,op(1,pp)):\np3 := plots[polygo
nplot]([seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2..25)],\n        \+
   style=patchnogrid,color=COLOR(RGB,.9,.9,.9)):\nplots[display]([p1,p
2,p3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 528 394 394 {PLOTDATA 2 "6)-%'CU
RVESG6%7S7$$!3;+++++++A!#<$!38++++++WA!#;7$$!3gLLL8#)\\g?F*$!3!Gl\"QfQ
'H2#F-7$$!3)ommYX=\"R>F*$!3c<i&=!GJF>F-7$$!3gLLL@Uh-=F*$!3)[)>;!=Lqw\"
F-7$$!3_LLL\\^?l;F*$!3k\"H.%Q\\X4;F-7$$!3'omm'e\"\\%G:F*$!3i<i**4]Pc9F
-7$$!3^LLL\\)e;S\"F*$!3A6\"eC$=z<8F-7$$!3C+++G[Pq7F*$!3A.;w#3&ox6F-7$$
!3dLLL02gM6F*$!3dc4KSCTO5F-7$$!3)>+++7?E***!#=$!3]!HEYm?E**)F*7$$!3tmm
m1`\\+')FY$!32(GL0W\"3?wF*7$$!3]MLL85JutFY$!3#4O!))RMDVkF*7$$!3M+++?z
\"R*fFY$!3arpJc[Sa^F*7$$!33.+++o&yg%FY$!3;lI*G))3')*QF*7$$!3%>+++3I@F$
FY$!3$)4Wg;CxCFF*7$$!3ONLL`A;f?FY$!3]'>AXHJ(*o\"F*7$$!3=-nmm=Eoh!#>$!3
OO\"*yZpls\\FY7$$\"3AILLLl@]gFgp$\"3GWkc1\"oN![FY7$$\"3^(*****f6KE?FY$
\"3ug`c`r**z:F*7$$\"3kFLL`?T%G$FY$\"3a\\eI<gl>DF*7$$\"3s)*****frtkYFY$
\"3k]6G+?>9NF*7$$\"3m)*****zI8zfFY$\"3(e\\;,91eU%F*7$$\"3CLLLLtc]tFY$
\"3S&*zWla9S`F*7$$\"3DJLL$4s*4')FY$\"3uxo=![hm9'F*7$$\"3=lmmm-To**FY$
\"3!)yu)3=O5)pF*7$$\"3KmmmuX%z8\"F*$\"3_I+KU(Q'3yF*7$$\"3w*****ziv2E\"
F*$\"3E/7%)R)\\m\\)F*7$$\"3]mmmIlV$R\"F*$\"3rwpL!)o#e?*F*7$$\"3^*****R
A)[I:F*$\"3-Bt>)e6:!**F*7$$\"3o*****zcmXm\"F*$\"3%*p'p%o]da5F-7$$\"3e*
*****fUH%z\"F*$\"3/0J&)=i[86F-7$$\"3h*****RuM$Q>F*$\"3IX-APO&\\<\"F-7$
$\"3OKLLD:wn?F*$\"3#Hy,xWXmA\"F-7$$\"3B+++SA&f?#F*$\"3O)*Q[j#R\"y7F-7$
$\"3?mmmmN<JBF*$\"3Q%zSN$=]@8F-7$$\"3!******>bt!oCF*$\"3M5=zN,Kl8F-7$$
\"3[KLLpd)of#F*$\"3`o+dXq7.9F-7$$\"3:+++_;`JFF*$\"3,:A,0))4R9F-7$$\"31
LLL8E<jGF*$\"3ijCH\\Bwq9F-7$$\"3<*******\\))4+$F*$\"3xunG-w>+:F-7$$\"3
,nmmy:sLJF*$\"3#HAt'ph&\\_\"F-7$$\"3KmmmA.YpKF*$\"3)3<%)z<Jma\"F-7$$\"
3fLLL\"3vSS$F*$\"3w778\\t[k:F-7$$\"3l*****R1sx_$F*$\"3#z6Yw2+xd\"F-7$$
\"3YLLLlFapOF*$\"3'fk0;!)z!*e\"F-7$$\"31mmmqTM'z$F*$\"39byJIC&ef\"F-7$
$\"3L+++;4aJRF*$\"37)y\"QN8`*f\"F-7$$\"3y*****>XT41%F*$\"3E8G%RhG'*f\"
F-7$$\"3;+++++++UF*$\"33++++++'f\"F--%'COLOURG6&%$RGBG$\"*++++\"!\")$
\"\"!Fb[lFa[l-%*THICKNESSG6#\"\"#-F$6%7S7$F($!3k+++++!o^#F-7$F/$!3aPb)
\\)=^[@F-7$F4$!3u\"H!QJw>d=F-7$F9$!3_q7lC,dg:F-7$F>$!3q**fOc&=OH\"F-7$
FC$!3[cuTVn\"z0\"F-7$FH$!3K!zoe=3xk)F*7$FM$!3]DUVEQv\"*oF*7$FR$!3u)\\H
`0\\DK&F*7$FW$!3/.eNv8O$*RF*7$Fgn$!3GiEd!\\@_&GF*7$F\\o$!3'yHsULKC.#F*
7$Fao$!3'fZO>ObJH\"F*7$Ffo$!3M\"4EpIf![tFY7$F[p$!3%o<0`:#RiNFY7$F`p$!3
7q(zrIc$f8FY7$Fep$!3-nv\"**H#*[;\"Fgp7$F[q$!3'*)4BFr1g2\"Fgp7$F`q$!35)
QxK%Hf[6FY7$Feq$!3MdDXDt!>)GFY7$Fjq$!32\"yV^i&*G^&FY7$F_r$!3+=D'Qqnue)
FY7$Fdr$!3rgOEvywB7F*7$Fir$!39f'e(3un&e\"F*7$F^s$!31]P_6J_!*>F*7$Fcs$!
3W=&)G<)37T#F*7$Fhs$!3)H#z#p6$fkFF*7$F]t$!3/>H2X'3%>JF*7$Fbt$!3+s@_me<
UMF*7$Fgt$!3$\\u#pFZ>+PF*7$F\\u$!3YnvU`)f<)QF*7$Fau$!3I0S9xm#)))RF*7$F
fu$!3EDhe\"*R\"f)RF*7$F[v$!3-muL(G:S'QF*7$F`v$!36+9*oV]Yj$F*7$Fev$!3+t
')>\"))p,C$F*7$Fjv$!3RkkCx!G&=FF*7$F_w$!3>sJ2LN6.?F*7$Fdw$!3Z='\\Msx;7
\"F*7$Fiw$\"3yLhQfPO-*)!#?7$F^x$\"3H%3Z(3T<88F*7$Fcx$\"39uNF08O!)GF*7$
Fhx$\"3#om%Q%z6Bo%F*7$F]y$\"3)R.)y%3;#olF*7$Fby$\"3Hnx&yjdd,*F*7$Fgy$
\"31nf**f%4x9\"F-7$F\\z$\"3UTMh&4%))R9F-7$Faz$\"3ke-7#fC'\\<F-7$Ffz$\"
3k+++++!o6#F--F[[l6&F][lFa[lFa[lF^[lFc[l-F$6%7$7$$!\"#Fb[lFa[l7$Fdel$F
bclFb[l-%&COLORG6&F][l$\"\"&!\"\"F[flF[fl-%*LINESTYLEG6#\"\"$-F$6%7$7$
$\"\"%Fb[lFa[l7$Fffl$\"#;Fb[lFhelF^fl-%)POLYGONSG6<7&Ffel7$$!+]TVQ<!\"
*$!+\\E'Hp\"F`[l7$F`gl$!+Pu-K9F`[lFfel7&F_gl7$$!+.r%3^\"Fbgl$!+zN%pV\"
F`[l7$Fjgl$!+fDnH5F`[lFegl7&Figl7$$!+l;!\\D\"Fbgl$!+^\"*Rh6F`[l7$Fchl$
!+Qj_+nFbglF^hl7&Fbhl7$$!+]lfs**!#5$!+fSgs*)Fbgl7$F\\il$!+J>QvRFbglFgh
l7&F[il7$$!+D<U3uF^il$!+0XevkFbgl7$Ffil$!+q/:`?FbglFail7&Feil7$$!+DM5J
]F^il$!+dF+yUFbgl7$F_jl$!+&[tq'))F^ilFjil7&F^jl7$$!+D!G&pDF^il$!+jrk@@
Fbgl7$Fhjl$!+3ZR]@F^ilFcjl7&Fgjl7$$!++DKwB!#7$!+\"pA;!>!#67$Fa[m$!+vWT
&p\"!#9F\\[m7&F`[m7$$\"+vs$Q^#F^il$\"+Ug(y%>Fbgl7$F]\\m$!+yW&pt\"F^ilF
g[m7&F\\\\m7$$\"++82C^F^il$\"+tfpOQFbgl7$Ff\\m$!+z.XJlF^ilFa\\m7&Fe\\m
7$$\"+]o;BuF^il$\"+>%*\\(Q&Fbgl7$F_]m$!+0/1W7FbglFj\\m7&F^]m7$$\"+!RS6
+\"Fbgl$\"+S5%o+(Fbgl7$Fh]m$!+p6U.?FbglFc]m7&Fg]m7$$\"+]o-h7Fbgl$\"+%3
E!)\\)Fbgl7$Fa^m$!+\"[&HlFFbglF\\^m7&F`^m7$$\"+5cZ6:Fbgl$\"+gjC2)*Fbgl
7$Fj^m$!+x5i+MFbglFe^m7&Fi^m7$$\"+xq!*Q<Fbgl$\"+!)eu)3\"F`[l7$Fc_m$!+N
+H8QFbglF^_m7&Fb_m7$$\"+!4X$4?Fbgl$\"+.$HP?\"F`[l7$F\\`m$!+#>P(**RFbgl
Fg_m7&F[`m7$$\"+g:WQAFbgl$\"+'=\"p*G\"F`[l7$Fe`m$!+@/)e\"QFbglF``m7&Fd
`m7$$\"+<_$\\]#Fbgl$\"+I\"ykP\"F`[l7$F^am$!+5KQ1JFbglFi`m7&F]am7$$\"+g
s#3u#Fbgl$\"+,%[9W\"F`[l7$Fgam$!+q.%p%>FbglFbam7&Ffam7$$\"+=#Q'**HFbgl
$\"+Iw#**\\\"F`[l7$F`bm$!+GsDbKFc[mF[bm7&F_bm7$$\"+`u3YKFbgl$\"+(ehJa
\"F`[l7$Fibm$\"+(4WIf#FbglFdbm7&Fhbm7$$\"+v8B.NFbgl$\"+$4A`d\"F`[l7$Fb
cm$\"+.D(f<'FbglF]cm7&Facm7$$\"+o(p$RPFbgl$\"+)=2Kf\"F`[l7$F[dm$\"+OB&
Q.\"F`[lFfcm7&FjcmFhflFhflF_dm-Fiel6&F][l$\"\"*F]flFedmFedm-%&STYLEG6#
%,PATCHNOGRIDG-%+AXESLABELSG6%Q\"x6\"Q!F_em-%%FONTG6#%(DEFAULTG-%%VIEW
G6$;$!#AF]fl$\"#UF]flFdem" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "f(x) =
 8*x-x^2;" "6#/-%\"fG6#%\"xG,&*&\"\")\"\"\"F'F+F+*$F'\"\"#!\"\"" }
{TEXT -1 2 " (" }{TEXT 260 3 "red" }{TEXT -1 6 ") and " }{XPPEDIT 18 
0 "g(x) = x^3-3*x^2;" "6#/-%\"gG6#%\"xG,&*$F'\"\"$\"\"\"*&F*F+*$F'\"\"
#F+!\"\"" }{TEXT -1 2 " (" }{TEXT 256 4 "blue" }{TEXT -1 26 "), the pi
cture shows that " }{XPPEDIT 18 0 "f(x) <= g(x);" "6#1-%\"fG6#%\"xG-%
\"gG6#F'" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "-2 <= x;" "6#1,$\"\"#!\"
\"%\"xG" }{XPPEDIT 18 0 "`` <= 0;" "6#1%!G\"\"!" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "g(x)<=f(x)" "6#1-%\"gG6#%\"xG-%\"fG6#F'" }{TEXT -1 5 " \+
for " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=4" "6#
1%!G\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 56 "The area of \+
the region enclosed between the curves from " }{XPPEDIT 18 0 "x = -2;
" "6#/%\"xG,$\"\"#!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 0;" "6
#/%\"xG\"\"!" }{TEXT -1 14 " is given by  " }{XPPEDIT 18 0 "Int(``(g(x
)-f(x)),x = -2 .. 0) = -Int(``(f(x)-g(x)),x = -2 .. 0);" "6#/-%$IntG6$
-%!G6#,&-%\"gG6#%\"xG\"\"\"-%\"fG6#F.!\"\"/F.;,$\"\"#F3\"\"!,$-F%6$-F(
6#,&-F16#F.F/-F,6#F.F3/F.;,$F7F3F8F3" }{TEXT -1 65 "., while the area \+
of the region enclosed between the curves from " }{XPPEDIT 18 0 "x=0" 
"6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=4" "6#/%\"xG\"\"%
" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "Int(``(f(x)-g(x)),x=0..4)" "6#-%$
IntG6$-%!G6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F-!\"\"/F-;\"\"!\"\"%" }
{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 5 "Now  " }{XPPEDIT 18 0 "f(x)-g(x) = 8*x+2*x^2-x^3;" "6#/,&-
%\"fG6#%\"xG\"\"\"-%\"gG6#F(!\"\",(*&\"\")F)F(F)F)*&\"\"#F)*$F(F2F)F)*
$F(\"\"$F-" }{TEXT -1 27 ", so the required area is: " }}{PARA 257 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-Int(``(8*x+2*x^2-x^3),x = -2 .. 0
)+Int(``(8*x+2*x^2-x^3),x = 0 .. 4);" "6#,&-%$IntG6$-%!G6#,(*&\"\")\"
\"\"%\"xGF-F-*&\"\"#F-*$F.F0F-F-*$F.\"\"$!\"\"/F.;,$F0F4\"\"!F4-F%6$-F
(6#,(*&F,F-F.F-F-*&F0F-*$F.F0F-F-*$F.F3F4/F.;F8\"\"%F-" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``=-``(4*x^2+2*x^3/3-x^4/4)" "6#/%!G,$-F$6#,(*&\"\"%\"
\"\"*$%\"xG\"\"#F+F+*(F.F+*$F-\"\"$F+F1!\"\"F+*&F-F*F*F2F2F2" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "PIECEWISE([0, ``],[``, ``],[-2, ``]);" "6#-%*
PIECEWISEG6%7$\"\"!%!G7$F(F(7$,$\"\"#!\"\"F(" }{XPPEDIT 18 0 " ``+ ``(
4*x^2+2*x^3/3-x^4/4)" "6#,&%!G\"\"\"-F$6#,(*&\"\"%F%*$%\"xG\"\"#F%F%*(
F-F%*$F,\"\"$F%F0!\"\"F%*&F,F*F*F1F1F%" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "PIECEWISE([4, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$\"\"%%!G7
$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-[0-(16-16/3-4)]+[
``(64+128/3-64)-0]" "6#/%!G,&7#,&\"\"!\"\"\",(\"#;F)*&F+F)\"\"$!\"\"F.
\"\"%F.F.F.7#,&-F$6#,(\"#kF)*&\"$G\"F)F-F.F)F5F.F)F(F.F)" }{TEXT -1 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "``=12-16/3+128/3" "6#/%!G,(\"#7\"\"\"*&\"#;F'\"\"$!
\"\"F+*&\"$G\"F'F*F+F'" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "``=12 + 112/3" "6#/%!G,&\"#7\"\"\"*&\"$7\"F'\"\"
$!\"\"F'" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``= 12 + 37" "6#/%!G,&\"#7\"\"\"\"#PF'" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "1/3;" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 1 " " }}
{PARA 257 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "``=49" "6#/%!G\"#\\
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/3 = 148/3" "6#/*&\"\"\"F%\"\"$!\"
\"*&\"$[\"F%F&F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "f := x -> 8*x-x^2: 'f(x)'=f
(x);\ng := x -> x^3-3*x^2; 'g(x)'=g(x);\n-Int('f(x)-g(x)',x=-2..0)+Int
('f(x)-g(x)',x=0..4);\n``=-Int(f(x)-g(x),x=-2..0)+Int(f(x)-g(x),x=0..4
);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"x
G,&*&\"\")\"\"\"F'F+F+*$)F'\"\"#F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"$\"\"\"F1*&F
0F1)F/\"\"#F1!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%
\"xG,&*$)F'\"\"$\"\"\"F,*&F+F,)F'\"\"#F,!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&-%$IntG6$,&-%\"fG6#%\"xG\"\"\"-%\"gGF*!\"\"/F+;!\"#\"
\"!F/-F%6$F'/F+;F3\"\"%F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-%$
IntG6$,(*&\"\")\"\"\"%\"xGF,F,*&\"\"#F,)F-F/F,F,*$)F-\"\"$F,!\"\"/F-;!
\"#\"\"!F4-F'6$F)/F-;F8\"\"%F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G
#\"$[\"\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 10 "Note that " }{XPPEDIT 18 0 "Int(f(x)-g(x),x=0..4)=128/3" 
"6#/-%$IntG6$,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F+!\"\"/F+;\"\"!\"\"%*&\"$G
\"F,\"\"$F0" }{TEXT -1 7 " while " }{XPPEDIT 18 0 "Int(g(x)-f(x),x=-2.
.0)=20/3" "6#/-%$IntG6$,&-%\"gG6#%\"xG\"\"\"-%\"fG6#F+!\"\"/F+;,$\"\"#
F0\"\"!*&\"#?F,\"\"$F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "Int('f(x)-g(x)',x=0..4
)=Int(f(x)-g(x),x=0..4);\n``=value(rhs(%));\nInt('g(x)-f(x)',x=-2..0)=
Int(g(x)-f(x),x=-2..0);\n``=value(rhs(%));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$,&-%\"fG6#%\"xG\"\"\"-%\"gGF*!\"\"/F+;\"\"!\"
\"%-F%6$,(*&\"\")F,F+F,F,*&\"\"#F,)F+F:F,F,*$)F+\"\"$F,F/F0" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/%!G#\"$G\"\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$,&-%\"gG6#%\"xG\"\"\"-%\"fGF*!\"\"/F+;!\"#\"
\"!-F%6$,(*$)F+\"\"$F,F,*&\"\"#F,)F+F;F,F/*&\"\")F,F+F,F/F0" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/%!G#\"#?\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 10 "Example 4 " }}{PARA 0 "" 0 "" {TEXT 296 8 "Ques
tion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 65 "Calculate the to
tal area of the region(s) bounded by the curves  " }{XPPEDIT 18 0 "y =
 x^3-4*x^2+3*x;" "6#/%\"yG,(*$%\"xG\"\"$\"\"\"*&\"\"%F)*$F'\"\"#F)!\"
\"*&F(F)F'F)F)" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "y = x^2-x;" "6#/
%\"yG,&*$%\"xG\"\"#\"\"\"F'!\"\"" }{TEXT -1 36 " between any points of
 intersection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 297 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 26 
"The two curves meet where " }{XPPEDIT 18 0 "x^3-4*x^2+3*x = x^2-x;" "
6#/,(*$%\"xG\"\"$\"\"\"*&\"\"%F(*$F&\"\"#F(!\"\"*&F'F(F&F(F(,&*$F&F,F(
F&F-" }{TEXT -1 17 ", that is, where " }{XPPEDIT 18 0 "x^3-5*x^2+4*x =
 0;" "6#/,(*$%\"xG\"\"$\"\"\"*&\"\"&F(*$F&\"\"#F(!\"\"*&\"\"%F(F&F(F(
\"\"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x*(x-1)*(x-4) = 0;" "6#/*(%
\"xG\"\"\",&F%F&F&!\"\"F&,&F%F&\"\"%F(F&\"\"!" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 58 "This shows that the curves meet at the th
ree points where " }{XPPEDIT 18 0 "x = 0,1;" "6$/%\"xG\"\"!\"\"\"" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 4;" "6#/%\"xG\"\"%" }{TEXT -1 
2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 105 "f := x -> x^3-4*x^2+3*x: 'f(x)'=f(x);\ng := x -> x^2
-x: 'g(x)'=g(x);\n'f(x)-g(x)'=f(x)-g(x);\nsolve(rhs(%));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*$)F'\"\"$\"\"\"F,*&\"\"%F,)F'\"
\"#F,!\"\"*&F+F,F'F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"
xG,&*$)F'\"\"#\"\"\"F,F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%
\"fG6#%\"xG\"\"\"-%\"gGF'!\"\",(*$)F(\"\"$F)F)*&\"\"&F))F(\"\"#F)F,*&
\"\"%F)F(F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!\"\"%\"\"\"" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The regio
n(s) can be illustrated as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 586 "f := x -> x^3-4*x^2+3*x:
\ng := x -> x^2-x:\na := 0: b := 4:\nc := -.5: d := 4.5:\np1 := plot([
f(x),g(x)],x=c..d,color=[red,blue],thickness=2):\np2 := plot([[a,g(a)]
,[a,f(a)]],color=COLOR(RGB,.4,.4,.4)):\np3 := plot([[b,0],[b,g(b)]],co
lor=COLOR(RGB,.5,.5,.5),linestyle=3):\npp := plot(f(x),x=a..b,adaptive
=false,numpoints=25):\nfpts := op(1,op(1,pp)):\npp := plot(g(x),x=a..b
,adaptive=false,numpoints=25):\ngpts := op(1,op(1,pp)):\np4 := plots[p
olygonplot]([seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2..25)],\n   \+
        style=patchnogrid,color=COLOR(RGB,.9,.9,.9)):\nplots[display](
[p1,p2,p3,p4]);" }}{PARA 13 "" 1 "" {GLPLOT2D 537 463 463 {PLOTDATA 2 
"6)-%'CURVESG6%7U7$$!3++++++++]!#=$!3+++++++DE!#<7$$!3rmmm\"HU,\"RF*$!
3\"oR)[%>%RW=F-7$$!3)HL$3FH'='HF*$!3A2&o:\\ZaE\"F-7$$!3mmm;/OU&*=F*$!3
!HRRtj=9>(F*7$$!3Gfmm\"H_\">#)!#>$!3mlFQ\\f^TFF*7$$\"3#fmm\"z%4\\Y#F@$
\"37W;M7\\>`rF@7$$\"3`LLeR-/P7F*$\"3*Qi,[LVz6$F*7$$\"3]***\\il'piAF*$
\"3)faY>'p,c[F*7$$\"3>MLe*)>VBLF*$\"3_MKS4cH>fF*7$$\"3Y++DJbw!Q%F*$\"3
#y-wDKrlI'F*7$$\"3UommTIOoaF*$\"3U*f*\\uc4zgF*7$$\"3eML$3_>jU'F*$\"3Hz
.=S,%QT&F*7$$\"3E,++D;v/vF*$\"3w%z?@a4D@%F*7$$\"3y+++v=h(e)F*$\"3qeu6U
t6(f#F*7$$\"3V+++v$[6j*F*$\"3A'ekipifB(F@7$$\"3UL$e*[z(y0\"F-$!35sBT?(
=\"*=\"F*7$$\"3wmm;a/cq6F-$!3+U06^-]_OF*7$$\"3%ommmJ<gE\"F-$!3G(oj3u](
ReF*7$$\"3/+]iSj0x8F-$!33.<bd]xE%)F*7$$\"3gmmm\"pW`Z\"F-$!3[\"f3<gO#p5
F-7$$\"3K+]i!f#=$e\"F-$!3ZymI)eD\"38F-7$$\"3?+](=xpeo\"F-$!3)y$eK^=^>:
F-7$$\"37nm\"H28Iz\"F-$!3nt<dfR>;<F-7$$\"3um;zpSS\"*=F-$!3kd0u(f)4p=F-
7$$\"3GLL3_?`(*>F-$!3K*=\"e&))>v*>F-7$$\"3fL$e*)>px5#F-$!3SRKTX\"*G$3#
F-7$$\"33+]Pf4t.AF-$!30+WmKAE7@F-7$$\"3uLLe*GstI#F-$!3/X8J1qP*3#F-7$$
\"30+++DRW9CF-$!3m-cIT*H(**>F-7$$\"3:++DJE>>DF-$!3_t0uvl6S=F-7$$\"3F+]
i!RU0i#F-$!3_&e#GTEW6;F-7$$\"3+++v=S2LFF-$!3%4r)QFoKk7F-7$$\"3Jmmm\"p)
=MGF-$!3%Q)e0^hf>')F*7$$\"3B++](=]@%HF-$!3f1t(>r%f0LF*7$$\"35L$e*[$z*R
IF-$\"3E0$=RO<$zCF*7$$\"3e++]iC$p9$F-$\"3q?HRWm7F**F*7$$\"3[m;H2qcZKF-
$\"3@&4-$)RAq!=F-7$$\"3O+]7.\"fFN$F-$\"3#*eGiFlk#y#F-7$$\"3Ymm;/OgbMF-
$\"3Qotk,m1mQF-7$$\"3w**\\ilAFjNF-$\"3[9OP\"RCZ9&F-7$$\"3yLLL$)*ppm$F-
$\"33gn)eflF_'F-7$$\"3)RL$3xe,tPF-$\"3G^BDs8y(3)F-7$$\"3Cn;HdO=yQF-$\"
30YZn*f&R-)*F-7$$\"3a+++D>#[(RF-$\"3)z5T(4sm_6!#;7$$\"3SnmT&G!e&3%F-$
\"3#=4o$4W_o8F`y7$$\"3#RLLL)Qk%=%F-$\"3_L8O#eF(y:F`y7$$\"37+]iSjE!H%F-
$\"3OpZ$>Jb8#=F`y7$$\"3L+++DM\"3M%F-$\"3@\\.:<sUW>F`y7$$\"3a+]P40O\"R%
F-$\"3q_5J.'4@2#F`y7$$\"3s+voa-oXWF-$\"3iz$z>y\\X@#F`y7$$\"3++++++++XF
-$\"3++++++]iBF`y-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F\\\\lF[\\l-%*T
HICKNESSG6#\"\"#-F$6%7S7$F($\"3++++++++vF*7$F/$\"3>#pudcj!RaF*7$F4$\"3
)yRmp7E\"RQF*7$F9$\"3]#))*3omoaAF*7$F>$\"3w_FgNpp%*))F@7$FD$!3:4=w\"p^
TS#F@7$FI$!3Y%\\[TQ8S3\"F*7$FN$!3m1&H//<2v\"F*7$FS$!3@]Zkq>\"*=AF*7$FX
$!3[!>rs'[lhCF*7$Fgn$!3[u+71O1yCF*7$F\\o$!3uw*[CEhlH#F*7$Fao$!3S&)fqH>
is=F*7$Ffo$!3$y*eV.T!H@\"F*7$F[p$!3))\\[tGZY_NF@7$F`p$\"3[#o&R$f!yAhF@
7$Fep$\"37j?#pA8l*>F*7$Fjp$\"3G0?L%HDyO$F*7$F_q$\"3)>;0jCyA>&F*7$Fdq$
\"3%yWBcnsH,(F*7$Fiq$\"3np3LY_%GB*F*7$F^r$\"3%['o%e6(Gc6F-7$Fcr$\"3R-M
t1G)=U\"F-7$Fhr$\"3e8GV&G0go\"F-7$F]s$\"37>p,ZAg#*>F-7$Fbs$\"3S@\">rz@
\\L#F-7$Fgs$\"3EeG$>=*p_EF-7$F\\t$\"3:-P:$f%f;IF-7$Fat$\"3z0%p>a&4:MF-
7$Fft$\"3l)H%4#))Qr#QF-7$F[u$\"3#3fP/.+nC%F-7$F`u$\"3Zvi@t&>mt%F-7$Feu
$\"3)Ra/$[oV)>&F-7$Fju$\"3U'G1$Qv49dF-7$F_v$\"3>(zMJ4&\\,iF-7$Fdv$\"3O
<j.h9DcnF-7$Fiv$\"3=&od&fW7*H(F-7$F^w$\"3ebDY+XB))yF-7$Fcw$\"3E))H'\\m
#f&[)F-7$Fhw$\"3R=tMtpjL\"*F-7$F]x$\"3OOVLv))pz(*F-7$Fbx$\"3ieR;$Hji/
\"F`y7$Fgx$\"3l]#RA[7i6\"F`y7$F\\y$\"3)32Y5uQC=\"F`y7$Fby$\"3ymoJ)fQ1E
\"F`y7$Fgy$\"3=v&)pa+mK8F`y7$F\\z$\"3+$[(G$*=h69F`y7$Ffz$\"3QEBRh'o#*[
\"F`y7$F`[l$\"3+++++++v:F`y-Fe[l6&Fg[lF[\\lF[\\lFh[lF]\\l-F$6$7$7$F[\\
lF[\\lF\\fl-%&COLORG6&Fg[l$\"\"%!\"\"F`flF`fl-F$6%7$7$$FaflF\\\\lF[\\l
7$Fgfl$\"#7F\\\\l-F^fl6&Fg[l$\"\"&FbflF]glF]gl-%*LINESTYLEG6#\"\"$-%)P
OLYGONSG6<7&F\\fl7$$\"+LBxV<!#5$\"+PQ/oSFjgl7$Fhgl$!+Q\")pR9FjglF\\fl7
&Fggl7$$\"+<$>5E$Fjgl$\"+0S9weFjgl7$Fbhl$!+=Yf(>#FjglF]hl7&Fahl7$$\"+L
AKn\\Fjgl$\"+M@!zD'Fjgl7$F[il$!+;K*)*\\#FjglFfhl7&Fjhl7$$\"+Lc$\\o'Fjg
l$\"+G&\\o;&Fjgl7$Fdil$!+\">*4;AFjglF_il7&Fcil7$$\"+<bQ%R)Fjgl$\"+tk.7
HFjgl7$F]jl$!+'p9yM\"FjglFhil7&F\\jl7$$\"+$Qk#z**Fjgl$\"+gZ\"G9%!#77$F
fjl$!+4?Ep?FjjlFajl7&Fejl7$$\"+l9.i6!\"*$!+*>J1Y$Fjgl7$F`[m$\"+2m&G)=F
jglF[[m7&F_[m7$$\"+=\"\\<L\"Fb[m$!+JCWqtFjgl7$Fj[m$\"+ef1=WFjglFe[m7&F
i[m7$$\"+&[A4]\"Fb[m$!+l^2F6Fb[m7$Fc\\m$\"+5#e%=vFjglF^\\m7&Fb\\m7$$\"
+(3Q\\n\"Fb[m$!+R`&z\\\"Fb[m7$F\\]m$\"+2&z/8\"Fb[mFg\\m7&F[]m7$$\"+B6@
G=Fb[m$!+?!eUx\"Fb[m7$Fe]m$\"+)yWT^\"Fb[mF`]m7&Fd]m7$$\"+g-w+?Fb[m$!+/
\"f2+#Fb[m7$F^^m$\"+e8G-?Fb[mFi]m7&F]^m7$$\"++z,u@Fb[m$!+\"z$=3@Fb[m7$
Fg^m$\"+IfL_DFb[mFb^m7&Ff^m7$$\"+SP)4M#Fb[m$!+pxzo?Fb[m7$F`_m$\"+J6ARJ
Fb[mF[_m7&F__m7$$\"+=Zg#\\#Fb[m$!+K2v()=Fb[m7$Fi_m$\"+kNZ?PFb[mFd_m7&F
h_m7$$\"+Fn*Gn#Fb[m$!+0eji9Fb[m7$Fb`m$\"+&=![rWFb[mF]`m7&Fa`m7$$\"+2xi
DGFb[m$!+w=2&**)Fjgl7$F[am$\"+IUae^Fb[mFf`m7&Fj`m7$$\"+X,H.IFb[m$\"+*3
.&z>!#67$Fdam$\"+]:Y;gFb[mF_am7&Fcam7$$\"+2:bgJFb[m$\"+t9L'4\"Fb[m7$F^
bm$\"+?V`GoFb[mFiam7&F]bm7$$\"+X@4LLFb[m$\"+a9E!f#Fb[m7$Fgbm$\"+-6TwxF
b[mFbbm7&Ffbm7$$\"+N;R(\\$Fb[m$\"+<fRWVFb[m7$F`cm$\"+9mNM()Fb[mF[cm7&F
_cm7$$\"+<4#)oOFb[m$\"+90r[lFb[m7$Ficm$\"+-gU\"z*Fb[mFdcm7&Fhcm7$$\"+7
lCEQFb[m$\"+a-'\\$*)Fb[m7$Fbdm$\"+'e\"R\"3\"Fj[lF]dm7&FadmFhflFhflFfdm
-F^fl6&Fg[l$\"\"*FbflF\\emF\\em-%&STYLEG6#%,PATCHNOGRIDG-%+AXESLABELSG
6%Q\"x6\"Q!Ffem-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!\"&Fbfl$\"#XFbflF[fm" 
1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 8 "Letting " }{XPPEDIT 18 0 "f(x) = x^3-4*x^2+3*x;" "6#/-%\"f
G6#%\"xG,(*$F'\"\"$\"\"\"*&\"\"%F+*$F'\"\"#F+!\"\"*&F*F+F'F+F+" }
{TEXT -1 2 " (" }{TEXT 260 3 "red" }{TEXT -1 6 ") and " }{XPPEDIT 18 
0 "g(x) = x^2-x;" "6#/-%\"gG6#%\"xG,&*$F'\"\"#\"\"\"F'!\"\"" }{TEXT 
-1 2 " (" }{TEXT 256 4 "blue" }{TEXT -1 26 "), the picture shows that \+
" }{XPPEDIT 18 0 "g(x) <= f(x);" "6#1-%\"gG6#%\"xG-%\"fG6#F'" }{TEXT 
-1 5 " for " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "
`` <= 1;" "6#1%!G\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)<=g(x
)" "6#1-%\"fG6#%\"xG-%\"gG6#F'" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "1<
=x" "6#1\"\"\"%\"xG" }{XPPEDIT 18 0 "``<=4" "6#1%!G\"\"%" }{TEXT -1 
68 ", so the area of the region enclosed between the curves is given b
y " }{XPPEDIT 18 0 "Int(``(f(x)-g(x)),x = 0 .. 1)+Int(``(g(x)-f(x)),x \+
= 1 .. 4) = Int(``(f(x)-g(x)),x = 0 .. 1)-Int(``(f(x)-g(x)),x = 1 .. 4
);" "6#/,&-%$IntG6$-%!G6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F/!\"\"/F/;\"\"!
F0F0-F&6$-F)6#,&-F26#F/F0-F-6#F/F4/F/;F0\"\"%F0,&-F&6$-F)6#,&-F-6#F/F0
-F26#F/F4/F/;F7F0F0-F&6$-F)6#,&-F-6#F/F0-F26#F/F4/F/;F0FCF4" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
5 "Now  " }{XPPEDIT 18 0 "f(x)-g(x) = x^3-5*x^2+4*x;" "6#/,&-%\"fG6#%
\"xG\"\"\"-%\"gG6#F(!\"\",(*$F(\"\"$F)*&\"\"&F)*$F(\"\"#F)F-*&\"\"%F)F
(F)F)" }{TEXT -1 27 ", so the required area is: " }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(x^3-5*x^2+4*x),x = 0 .. 1)-Int(`
`(x^3-5*x^2+4*x),x = 1 .. 4);" "6#,&-%$IntG6$-%!G6#,(*$%\"xG\"\"$\"\"
\"*&\"\"&F.*$F,\"\"#F.!\"\"*&\"\"%F.F,F.F./F,;\"\"!F.F.-F%6$-F(6#,(*$F
,F-F.*&F0F.*$F,F2F.F3*&F5F.F,F.F./F,;F.F5F3" }{TEXT -1 1 " " }}{PARA 
257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x^4/4-5*x^3/3+2*x^2" "6#
/%!G,(*&%\"xG\"\"%F(!\"\"\"\"\"*(\"\"&F**$F'\"\"$F*F.F)F)*&\"\"#F**$F'
F0F*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([1, ``],[0, ``])" "6
#-%*PIECEWISEG6$7$\"\"\"%!G7$\"\"!F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
- x^4/4-5*x^3/3+2*x^2" "6#,(*&%\"xG\"\"%F&!\"\"F'*(\"\"&\"\"\"*$F%\"\"
$F*F,F'F'*&\"\"#F**$F%F.F*F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWIS
E([4, ``],[1, ``])" "6#-%*PIECEWISEG6$7$\"\"%%!G7$\"\"\"F(" }{TEXT -1 
1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "``=[``(1/4-5/3+2)-0]-[``( 64-320/3+32)-(1/4-5/3+2)]
" "6#/%!G,&7#,&-F$6#,(*&\"\"\"F,\"\"%!\"\"F,*&\"\"&F,\"\"$F.F.\"\"#F,F
,\"\"!F.F,7#,&-F$6#,(\"#kF,*&\"$?$F,F1F.F.\"#KF,F,,(*&F,F,F-F.F,*&F0F,
F1F.F.F2F,F.F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(1/4-5/3+2)-``
(96-320/3)+``(1/4-5/3+2);" "6#/%!G,(-F$6#,(*&\"\"\"F*\"\"%!\"\"F**&\"
\"&F*\"\"$F,F,\"\"#F*F*-F$6#,&\"#'*F**&\"$?$F*F/F,F,F,-F$6#,(*&F*F*F+F
,F**&F.F*F/F,F,F0F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(1/2-10/3+4)
-(96-320/3)" "6#/%!G,&-F$6#,(*&\"\"\"F*\"\"#!\"\"F**&\"#5F*\"\"$F,F,\"
\"%F*F*,&\"#'*F**&\"$?$F*F/F,F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/
2+310/3-92" "6#/%!G,(*&\"\"\"F'\"\"#!\"\"F'*&\"$5$F'\"\"$F)F'\"##*F)" 
}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``=103" "6#/%!G\"$.\"" }{TEXT -1 1 " " 
}{XPPEDIT 18 0 "1/3 -92 +1/2" "6#,(*&\"\"\"F%\"\"$!\"\"F%\"##*F'*&F%F%
\"\"#F'F%" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``=11" "6#/%!G\"#6" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/3
 + 1/2" "6#,&*&\"\"\"F%\"\"$!\"\"F%*&F%F%\"\"#F'F%" }{TEXT -1 1 " " }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=11" "6#/%!G\"#6" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "5/6 = 71/6" "6#/*&\"\"&\"\"\"\"\"'!\"\"
*&\"#rF&F'F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
182 "f := x -> x^3-4*x^2+3*x: 'f(x)'=f(x);\ng := x -> x^2-x: 'g(x)'=g(
x);\nInt('f(x)-g(x)',x=0..1)-Int('f(x)-g(x)',x=1..4);\n``=Int(f(x)-g(x
),x=0..1)-Int(f(x)-g(x),x=1..4);\n``=value(rhs(%));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*$)F'\"\"$\"\"\"F,*&\"\"%F,)F'\"\"#F,
!\"\"*&F+F,F'F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&*
$)F'\"\"#\"\"\"F,F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6
$,&-%\"fG6#%\"xG\"\"\"-%\"gGF*!\"\"/F+;\"\"!F,F,-F%6$F'/F+;F,\"\"%F/" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&-%$IntG6$,(*$)%\"xG\"\"$\"\"\"
F.*&\"\"&F.)F,\"\"#F.!\"\"*&\"\"%F.F,F.F./F,;\"\"!F.F.-F'6$F)/F,;F.F5F
3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"#r\"\"'" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{PARA 0 "" 0 "" 
{TEXT 306 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 62 
"Calculate the total area of the region bounded by the curves  " }
{XPPEDIT 18 0 "y = sin*x;" "6#/%\"yG*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 
7 "  and  " }{XPPEDIT 18 0 "y = cos*x;" "6#/%\"yG*&%$cosG\"\"\"%\"xGF'
" }{TEXT -1 78 " between their first two points of intersection which \+
lie to the right of the " }{TEXT 308 1 "y" }{TEXT -1 6 " axis." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 307 8 "Solution
" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 42 "The region can be illustrated as follows. " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 562 "f := x
 -> sin(x): \ng := x -> cos(x):\na := Pi/4: b := 5*Pi/4:\nc := 0: d :=
 2*Pi:\np1 := plot([f(x),g(x)],x=c..d,color=[red,blue],thickness=2):\n
p2 := plot([[[a,0],[a,g(a)]],[[b,0],[b,g(b)]]],\n                color
=COLOR(RGB,.5,.5,.5),linestyle=3):\npp := plot(f(x),x=a..b,adaptive=fa
lse,numpoints=25):\nfpts := op(1,op(1,pp)):\npp := plot(g(x),x=a..b,ad
aptive=false,numpoints=25):\ngpts := op(1,op(1,pp)):\np3 := plots[poly
gonplot]([seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2..25)],\n      \+
     style=patchnogrid,color=COLOR(RGB,.9,.9,.9)):\nplots[display]([p1
,p2,p3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 625 228 228 {PLOTDATA 2 "6)-%'
CURVESG6%7en7$$\"\"!F)F(7$$\"3i]cC&eb&p8!#=$\"3+9&o'z\"y_O\"F-7$$\"3Yq
R*pd)>hDF-$\"3K!y=V&*)GLDF-7$$\"3Zs[E^dK,RF-$\"3R]F<<.6.QF-7$$\"3ab^Ne
hL]_F-$\"3I,*y(H4U7]F-7$$\"35*QhW&\\$Hf'F-$\"3%=pD*eceDhF-7$$\"3zS11.f
pPyF-$\"3w>82*oU&fqF-7$$\"3upr'fwtl7*F-$\"3@)*QjP!>8\"zF-7$$\"3!QRa&4L
&f/\"!#<$\"3[[%3w7ESl)F-7$$\"3YjXwg<#)y6FQ$\"3Pi.Z,bcT#*F-7$$\"36qGW%H
$\\:8FQ$\"3!e$pSF\"oen*F-7$$\"3SH22vMov8FQ$\"35_*\\T'zD5)*F-7$$\"3$*)e
)pbO(eV\"FQ$\"3#ffI'ft64**F-7$$\"3/]+3VNj.:FQ$\"3)\\\"3MzUXx**F-7$$\"3
96:YIMRr:FQ$\"2-6Ot@)******FQ7$$\"3Z'z]kaJ%R;FQ$\"3Ay8:y_Xw**F-7$$\"3!
=3SCmpuq\"FQ$\"3s;Nd#HZn!**F-7$$\"33u?N%*p.t<FQ$\"3'))pQ.m*='z*F-7$$\"
3LmSEEVgQ=FQ$\"3*p]U&fD`V'*F-7$$\"3S`:%R;(od>FQ$\"3cBCb^l'3E*F-7$$\"3#
=uye<)G*4#FQ$\"3;2*>c4&oN')F-7$$\"3Ua[#o!GC>AFQ$\"3Se*Rp1I-(zF-7$$\"3-
YwJf&y(eBFQ$\"3c:s(f4sF0(F-7$$\"3oNrpV8H#[#FQ$\"3(y]$4EukDhF-7$$\"3EzY
)QW/yh#FQ$\"3?Pawd/k,]F-7$$\"35v)>8'\\%ou#FQ$\"3i\"*R%R(HvXQF-7$$\"3w$
GCS?&[\")GFQ$\"3!\\o6f)Q%=d#F-7$$\"3Z%)='p(p70IFQ$\"3Kh5Xo]Ug8F-7$$\"3
sA%)\\K8\\QJFQ$\"3^)pb)>hJ,J!#?7$$\"3c&fJlT>qF$FQ$!3)[9aqyJ,N\"F-7$$\"
39l2\"4_3wR$FQ$!3w\"*=GeHGKDF-7$$\"3MOnGd![y_$FQ$!39*)zsmMAnPF-7$$\"3#
*=4JU#)RiOFQ$!3#35.)=3zv\\F-7$$\"3%G,f%[$HSz$FQ$!3!fgI[?W72'F-7$$\"3Ko
E+7#*Q@RFQ$!3#=$4p_xMJqF-7$$\"3y(f0ii+G1%FQ$!3WWqZN&GL'zF-7$$\"3;fORr]
')*=%FQ$!33*3&3nLil')F-7$$\"3R&p%*z[LbK%FQ$!3]$[$Q0***4E*F-7$$\"3M'pEx
Bp%[WFQ$!3'R4g7n[Pl*F-7$$\"3,z&[2')pc^%FQ$!30-Py@682)*F-7$$\"3oh/x$[qG
e%FQ$!3$euj/)>C;**F-7$$\"3qQq'H.,hk%FQ$!3Ief(ReP!y**F-7$$\"3e;O;#eJ$4Z
FQ$!3;'*phhK&*****F-7$$\"3sU'G^sDax%FQ$!311%\\AUQ,)**F-7$$\"3&)oO4o)>:
%[FQ$!32^6Oe=u;**F-7$$\"3;@9vt*Qh!\\FQ$!31K)*)[7\"*G\")*F-7$$\"3Ou\"4%
z!e2(\\FQ$!3_#49llz!o'*F-7$$\"3*zX#[4&eg5&FQ$!3]_&yQBx]B*F-7$$\"3n*e![
/*ojB&FQ$!3o\"*oMmwMe')F-7$$\"3#ob8X5I'p`FQ$!3Froqht!o\"zF-7$$\"3PA\"G
X$yy,bFQ$!3x5TwV@sUqF-7$$\"39L1/jqABcFQ$!3)\\'4LY'Q38'F-7$$\"3mweKB,Ti
dFQ$!3iCF-zq_v\\F-7$$\"3'oIe;6(*o)eFQ$!3s)y_Bpo*fQF-7$$\"3nKcu0ii>gFQ$
!3dChVjR=0EF-7$$\"3w)*zj%)[mYhFQ$!3]'*=#eVn4O\"F-7$$\"3)****>YH&=$G'FQ
$!3/UE[]'efD\"!#D-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"
\"#-F$6%7en7$F($\"\"\"F)7$$\"3;`#Gi#zxZo!#>$\"3!eFW#HJcw**F-7$F+$\"3#*
p#fVPij!**F-7$$\"3q5)>63x`'>F-$\"3wuHGMb[2)*F-7$F1$\"3[#y/-5-Qn*F-7$F6
$\"3'4g??['e[#*F-7$F;$\"35D5SC42`')F-7$F@$\"3FQ^ZT?D/zF-7$FE$\"3=_>U!=
uD3(F-7$FJ$\"3')e\\phdX;hF-7$FO$\"3[yHev:x5]F-7$FU$\"39g)pymR,#QF-7$FZ
$\"39.oK.jQDDF-7$F^o$\"3KlS\"H&o8X8F-7$Fho$!3)R^L8JO5(f!#@7$Fbp$!33@,A
BB[i8F-7$F\\q$!3!p()*f7@=YEF-7$Faq$!3%R#pX#*)3Jx$F-7$Ffq$!3nht,N_JU]F-
7$F[r$!3L_)*oQ%*[RgF-7$F`r$!3k\\\"Q8J;$*3(F-7$Fer$!3UtA<tT?/zF-7$Fjr$!
3%>:G>r1$f')F-7$F_s$!3(H:T1DO4B*F-7$Fds$!3E3Gi$\\BOm*F-7$$\"3*Q3$\\!41
L%HFQ$!33D)eOYbS!)*F-7$Fis$!3Zw5r5+.2**F-7$$\"3g`,ta\"4=2$FQ$!3iPR!e>h
c(**F-7$F^t$!3'p)f24>&*****F-7$$\"3#*3]^u`v2KFQ$!3Ms?.b/7y**F-7$Fdt$!3
8)zW\"G!Q%3**F-7$$\"3e!=@(oRJPLFQ$!3\\.t5wk24)*F-7$Fit$!3;&3vb[lSn*F-7
$F^u$!3eDXLTAEj#*F-7$Fcu$!35CZ!4<'=u')F-7$Fhu$!37)oJ.#y1YzF-7$F]v$!3e'
42e*fc5rF-7$Fbv$!3k9U,*\\'e[gF-7$Fgv$!3![Trb\\)o!*\\F-7$F\\w$!3'QYDqc
\"ysPF-7$Faw$!3L'GUFnl'3EF-7$F[x$!3=4d,soc\"H\"F-7$Fex$!35e,\"yX$RdIFb
t7$F_y$\"3o*y!R^Js(G\"F-7$Fiy$\"3/CaOV7/bDF-7$F^z$\"3(p'>9&G)zNQF-7$Fc
z$\"3!p+]&z/I.]F-7$Fhz$\"32-G'*3.N4hF-7$F][l$\"3EO\\&RI+$*4(F-7$Fb[l$
\"3KytI?$y,!zF-7$Fg[l$\"3C%Qb^XPVn)F-7$F\\\\l$\"3`bo#[!4+D#*F-7$Fa\\l$
\"3Yfw#e$))oa'*F-7$$\"3G:=>Xb9$3'FQ$\"3**=**yvne+)*F-7$Ff\\l$\"3n&\\^
\"=b&p!**F-7$$\"3P***G'*3D\\@'FQ$\"37jT:e<rw**F-7$F[]l$\"2))**********
*****FQ-Fa]l6&Fc]lF(F(Fd]lFg]l-F$6%7$7$$\"3!G[uRj\")R&yF-F(7$Fajl$\"3s
va'=\"y1rqF--%&COLORG6&Fc]l$\"\"&!\"\"FijlFijl-%*LINESTYLEG6#\"\"$-F$6
%7$7$$\"3RTs)p\"3*p#RFQF(7$Fd[m$!3sva'=\"y1rqF-FfjlF\\[m-%)POLYGONSG6<
7&7$$\"+];)R&y!#5$\"+By1rqF`\\m7$$\"+Ms`B#*F`\\m$\"+![`-(zF`\\m7$Fd\\m
$\"+T&e%RgF`\\m7$F^\\m$\"+,y1rqF`\\m7&Fc\\m7$$\"+A!=:/\"!\"*$\"+qorJ')
F`\\m7$F`]m$\"+b_5\\]F`\\mFh\\m7&F_]m7$$\"+S2`v6Fb]m$\"+<L%*G#*F`\\m7$
Fj]m$\"+7J`]QF`\\mFe]m7&Fi]m7$$\"+!yJ/J\"Fb]m$\"+h>'Hm*F`\\m7$Fc^m$\"+
*GGVd#F`\\mF^^m7&Fb^m7$$\"+g;pW9Fb]m$\"+yLf?**F`\\m7$F\\_m$\"+_qqd7F`
\\mFg^m7&F[_m7$$\"+ax;p:Fb]m$\"+Rn)*****F`\\m7$Fe_m$\"+%er&G;!#7F`_m7&
Fd_m7$$\"+Tb0)p\"Fb]m$\"+&oM\">**F`\\m7$F_`m$!+++;p7F`\\mFi_m7&F^`m7$$
\"+t9NJ=Fb]m$\"+=6Zi'*F`\\m7$Fh`m$!+/+<wDF`\\mFc`m7&Fg`m7$$\"+D*>U'>Fb
]m$\"+)H?gB*F`\\m7$Faam$!+up_LQF`\\mF\\am7&F`am7$$\"+e9*35#Fb]m$\"+Y(*
eF')F`\\m7$Fjam$!+*=ah0&F`\\mFeam7&Fiam7$$\"+>=F@AFb]m$\"+q%fz&zF`\\m7
$Fcbm$!+\"p[c0'F`\\mF^bm7&Fbbm7$$\"+%f\"zcBFb]m$\"+\"RWo1(F`\\m7$F\\cm
$!+7()GvqF`\\mFgbm7&F[cm7$$\"+Ey'G\\#Fb]m$\"+,xqTgF`\\m7$Fecm$!+z'[&oz
F`\\mF`cm7&Fdcm7$$\"+*[-Si#Fb]m$\"+aR(y%\\F`\\m7$F^dm$!+#eS,p)F`\\mFic
m7&F]dm7$$\"+F`3VFFb]m$\"+e1V!)QF`\\m7$Fgdm$!+@CT;#*F`\\mFbdm7&Ffdm7$$
\"+Rjo%)GFb]m$\"+,i*3a#F`\\m7$F`em$!+'y1=n*F`\\mF[em7&F_em7$$\"+q4k/IF
b]m$\"+(zR_O\"F`\\m7$Fiem$!+kwO1**F`\\mFdem7&Fhem7$$\"+An<WJFb]m$!+ka1
%e#F\\`m7$Fbfm$!+8m'*****F`\\mF]fm7&Fafm7$$\"+1&*onKFb]m$!+G&HwD\"F`\\
m7$F[gm$!+1Kg?**F`\\mFffm7&Fjfm7$$\"+1E?.MFb]m$!+-1O'e#F`\\m7$Fdgm$!+%
Q[(f'*F`\\mF_gm7&Fcgm7$$\"+CJCKNFb]m$!+-'**y!QF`\\m7$F]hm$!+JchY#*F`\\
mFhgm7&F\\hm7$$\"+mL)om$Fb]m$!+Sak9]F`\\m7$Ffhm$!+i>y^')F`\\mFahm7&Feh
m7$$\"+R^_!z$Fb]m$!+*HiL/'F`\\m7$F_im$!+-RHnzF`\\mFjhm7&F^im7$$\"+43*p
#RFb]m$!+cx1rqF`\\m7$Fhim$!+ny1rqF`\\mFcim-Fgjl6&Fc]l$\"\"*F[[mFajmFaj
m-%&STYLEG6#%,PATCHNOGRIDG-%+AXESLABELSG6%Q\"x6\"Q!F[[n-%%FONTG6#%(DEF
AULTG-%%VIEWG6$;F($\"+3`=$G'Fb]mF`[n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The graph of  " }
{XPPEDIT 18 0 "y = cos*x;" "6#/%\"yG*&%$cosG\"\"\"%\"xGF'" }{TEXT -1 
13 " is drawn in " }{TEXT 256 4 "blue" }{TEXT -1 22 ", while the graph
 of  " }{XPPEDIT 18 0 "y = sin*x;" "6#/%\"yG*&%$sinG\"\"\"%\"xGF'" }
{TEXT -1 13 " is drawn in " }{TEXT 260 3 "red" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 51 "The first two points of intersection of t
he curves " }{XPPEDIT 18 0 "y = sin*x;" "6#/%\"yG*&%$sinG\"\"\"%\"xGF'
" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = cos*x;" "6#/%\"yG*&%$cosG\"
\"\"%\"xGF'" }{TEXT -1 13 " occur where " }{XPPEDIT 18 0 "x=Pi/4" "6#/
%\"xG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 11 " and where " }{XPPEDIT 18 
0 "x=5*Pi/4" "6#/%\"xG*(\"\"&\"\"\"%#PiGF'\"\"%!\"\"" }{TEXT -1 2 ". \+
" }}{PARA 0 "" 0 "" {TEXT -1 23 "The picture shows that " }{XPPEDIT 
18 0 "cos*x <= sin*x;" "6#1*&%$cosG\"\"\"%\"xGF&*&%$sinGF&F'F&" }
{TEXT -1 5 " for " }{XPPEDIT 18 0 "Pi/4 <= x;" "6#1*&%#PiG\"\"\"\"\"%!
\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 5*Pi/4;" "6#1%!G*(\"\"&\"\"\"%#PiGF'
\"\"%!\"\"" }{TEXT -1 79 ", so the area of the region enclosed between
 the curves over the interval from " }{XPPEDIT 18 0 "x=Pi/4" "6#/%\"xG
*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=5*Pi/4" 
"6#/%\"xG*(\"\"&\"\"\"%#PiGF'\"\"%!\"\"" }{TEXT -1 5 " is: " }}{PARA 
257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(sin*x-cos*x),x = Pi/
4 .. 5*Pi/4) = -cos*x-sin*x;" "6#/-%$IntG6$-%!G6#,&*&%$sinG\"\"\"%\"xG
F-F-*&%$cosGF-F.F-!\"\"/F.;*&%#PiGF-\"\"%F1*(\"\"&F-F5F-F6F1,&*&F0F-F.
F-F1*&F,F-F.F-F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([5*Pi/4, `
`],[Pi/4, ``])" "6#-%*PIECEWISEG6$7$*(\"\"&\"\"\"%#PiGF)\"\"%!\"\"%!G7
$*&F*F)F+F,F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(-cos(5*Pi/4)-si
n(5*Pi/4))-(-cos(Pi/4)-sin(Pi/4))" "6#/%!G,&-F$6#,&-%$cosG6#*(\"\"&\"
\"\"%#PiGF.\"\"%!\"\"F1-%$sinG6#*(F-F.F/F.F0F1F1F.,&-F*6#*&F/F.F0F1F1-
F36#*&F/F.F0F1F1F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(1/sqrt(2)+1/sqr
t(2))-(-1/sqrt(2)-1/sqrt(2))" "6#/%!G,&-F$6#,&*&\"\"\"F*-%%sqrtG6#\"\"
#!\"\"F**&F*F*-F,6#F.F/F*F*,&*&F*F*-F,6#F.F/F/*&F*F*-F,6#F.F/F/F/" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``=4/sqrt(2)" "6#/%!G*&\"\"%\"\"\"-%%sq
rtG6#\"\"#!\"\"" }{XPPEDIT 18 0 "``=2*sqrt(2)" "6#/%!G*&\"\"#\"\"\"-%%
sqrtG6#F&F'" }{TEXT -1 1 " " }{TEXT 309 1 "~" }{TEXT -1 15 " 2.8284271
25.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 76 "Int(sin(x)-cos(x),x=Pi/4..5*Pi/4);\n``=value(%);\n``=
evalf(evalf[13](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$
,&-%$sinG6#%\"xG\"\"\"-%$cosGF)!\"\"/F*;,$*&\"\"%F.%#PiGF+F+,$*(\"\"&F
+F3F.F4F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"#\"\"\"F'#F
(F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+DrUGG!\"*" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 6 " }}{PARA 0 "" 0 "
" {TEXT 290 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 
65 "Calculate the area of the region enclosed between the graphs of  \+
" }{XPPEDIT 18 0 "y = sin*x;" "6#/%\"yG*&%$sinG\"\"\"%\"xGF'" }{TEXT 
-1 6 " and  " }{XPPEDIT 18 0 "y = x;" "6#/%\"yG%\"xG" }{TEXT -1 6 " fr
om " }{XPPEDIT 18 0 "x = -Pi/2;" "6#/%\"xG,$*&%#PiG\"\"\"\"\"#!\"\"F*
" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = Pi/2;" "6#/%\"xG*&%#PiG\"\"\"
\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT 291 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The region can be illustr
ated as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 661 "f := x -> x:\ng := x -> sin(x):\na := -Pi/2:
 b := Pi/2:\nc := -1.75: d := 1.75:\np1 := plot([f(x),g(x)],x=c..d,col
or=[red,blue],thickness=2):\np2 := plot([[[a,g(a)],[a,f(a)]],[[b,g(b)]
,[b,f(b)]]],\n                 color=COLOR(RGB,.4,.4,.4)):\np3 := plot
([[[a,0],[a,g(a)]],[[b,0],[b,g(b)]]],\n                color=COLOR(RGB
,.5,.5,.5),linestyle=3):\npp := plot(f(x),x=a..b,adaptive=false,numpoi
nts=25):\nfpts := op(1,op(1,pp)):\npp := plot(g(x),x=a..b,adaptive=fal
se,numpoints=25):\ngpts := op(1,op(1,pp)):\np4 := plots[polygonplot]([
seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2..25)],\n           style
=patchnogrid,color=COLOR(RGB,.9,.9,.9)):\nplots[display]([p1,p2,p3,p4]
);" }}{PARA 13 "" 1 "" {GLPLOT2D 437 315 315 {PLOTDATA 2 "6+-%'CURVESG
6%7S7$$!3+++++++]<!#<F(7$$!3ummTg*4Pn\"F*F,7$$!3BLe*[SItg\"F*F/7$$!3em
;H_'zE`\"F*F27$$!3om;/mS`d9F*F57$$!3CLekLcu#Q\"F*F87$$!3sm\"HK=2MJ\"F*
F;7$$!3&)*\\iSB6;C\"F*F>7$$!3em\"H2wft;\"F*FA7$$!35+D\"GTYL4\"F*FD7$$!
37LL3(e9s,\"F*FG7$$!3-mmTNjd,&*!#=FJ7$$!3Y)***\\iQnY()FLFN7$$!3m****\\
(or'))zFLFQ7$$!3E****\\Ph>esFLFT7$$!3Cm;HdV&[f'FLFW7$$!3VKL$3#o21eFLFZ
7$$!3qJLL$yyy8&FLFgn7$$!3[**\\i:cggVFLFjn7$$!3gLLLeresOFLF]o7$$!38)*\\
il=s<HFLF`o7$$!3E(*\\(of6*)>#FLFco7$$!3_JLe*[3*[9FLFfo7$$!3R=Le9^r,w!#
>Fio7$$!3,*pm;ajvs\"!#?F]p7$$\"3'zL$3FR%Qa(F[pFap7$$\"3)4+Dcr;hU\"FLFd
p7$$\"3#[L$3Fgg^@FLFgp7$$\"3))*****\\Z26!HFLFjp7$$\"3#>+](=%[Vj$FLF]q7
$$\"3!G+vVt'zVVFLF`q7$$\"36***\\78=:8&FLFcq7$$\"3@kmmT3KReFLFfq7$$\"3K
/+]780&f'FLFiq7$$\"3uJ$3FWb)zsFLF\\r7$$\"3'\\++vBF&G!)FLF_r7$$\"3xl;/^
!pHt)FLFbr7$$\"3?,](=s8$p%*FLFer7$$\"3umm\"H_A*=5F*Fhr7$$\"3%)*\\Pfe!H
%4\"F*F[s7$$\"3#RLL$))*yo;\"F*F^s7$$\"3^L$eR666C\"F*Fas7$$\"3<nT5g&GZJ
\"F*Fds7$$\"3Y++]Z`P#Q\"F*Fgs7$$\"3#pm\"z*>1*f9F*Fjs7$$\"3[LLL=2DH:F*F
]t7$$\"33+vVQk=.;F*F`t7$$\"3I+DccB&Rn\"F*Fct7$$\"3+++++++]<F*Fft-%'COL
OURG6&%$RGBG$\"*++++\"!\")$\"\"!F`uF_u-%*THICKNESSG6#\"\"#-F$6%7S7$F($
!3Bp$R(o%f)R)*FL7$F,$!3iFGoU14Z**FL7$F/$!3O5o&z/FL***FL7$F2$!3%>m2^ZOF
***FL7$F5$!3T0EUTo#f$**FL7$F8$!3WF39J^qB)*FL7$F;$!3#elP0%)y0n*FL7$F>$!
3m,/(R%41j%*FL7$FA$!3yk&H#fH<(>*FL7$FD$!3Qy_'39'p\")))FL7$FG$!36Z/0!Gp
k])FL7$FJ$!3@R][u=2N\")FL7$FN$!3Yg#QTb-Ln(FL7$FQ$!3G'H?kzjc;(FL7$FT$!3
3Hjw-q[PmFL7$FW$!3KnPX!p-r7'FL7$FZ$!3WZ')R*Q@`[&FL7$Fgn$!3v8u.oez9\\FL
7$Fjn$!3qh8'pb>PA%FL7$F]o$!3a?TrAIe!f$FL7$F`o$!3G2f)4i*\\wGFL7$Fco$!3#
HG^![SB\"=#FL7$Ffo$!3C;k\\KU%QW\"FL7$Fio$!3yu)z&3gR%f(F[p7$F]p$!3c**[N
#oivs\"F_p7$Fap$\"3OL+_z1pOvF[p7$Fdp$\"3Z2#Gwa(G@9FL7$Fgp$\"3Ic`.1L/N@
FL7$Fjp$\"3S'=YYY$egGFL7$F]q$\"3O$*[mk!o[b$FL7$F`q$\"3?@LYZwZ3UFL7$Fcq
$\"3peJm^aD4\\FL7$Ffq$\"3fxA'GL(38bFL7$Fiq$\"378jYW$es7'FL7$F\\r$\"3G#
oi3irOl'FL7$F_r$\"3Wqg[5oS$>(FL7$Fbr$\"3%RBV962Xm(FL7$Fer$\"3ykGL*3ni6
)FL7$Fhr$\"39Z_4EhV:&)FL7$F[s$\"3103r03.')))FL7$F^s$\"3!enSPZ%G&>*FL7$
Fas$\"3]9KS7IWh%*FL7$Fds$\"3=)zc'[S$Rn*FL7$Fgs$\"3^uR!)GA,B)*FL7$Fjs$
\"39D,rv)z&Q**FL7$F]t$\"3I)4_L0r8***FL7$F`t$\"3U*fF>'[v%***FL7$Fct$\"3
/S2*=LTo%**FL7$Fft$\"3Bp$R(o%f)R)*FL-Fit6&F[uF_uF_uF\\uFau-F$6$7$7$$!3
c'*[zEjzq:F*$!\"\"F`u7$Fa_lFa_l-%&COLORG6&F[u$\"\"%Fd_lFi_lFi_l-F$6$7$
7$$\"3c'*[zEjzq:F*$\"\"\"F`u7$F_`lF_`lFf_l-F$6%7$7$Fa_lF_uF`_l-Fg_l6&F
[u$\"\"&Fd_lFj`lFj`l-%*LINESTYLEG6#\"\"$-F$6%7$7$F_`lF_uF^`lFh`lF\\al-
%)POLYGONSG6<7&7$$!+Cjzq:!\"*Fial7$$!+l2%QV\"F[blF]bl7$F]bl$!+qBO1**!#
57$FialFc_l7&F\\bl7$$!+mkn98F[blFfbl7$Ffbl$!+#4-Qn*FbblF_bl7&Febl7$$!+
\\Pm!=\"F[blF]cl7$F]cl$!+qke[#*FbblFhbl7&F\\cl7$$!+3FwX5F[blFdcl7$Fdcl
$!+442`')FbblF_cl7&Fccl7$$!+#GG]6*FbblF[dl7$F[dl$!+A?D/zFbblFfcl7&Fjcl
7$$!+LtEqyFbblFbdl7$Fbdl$!+eTd#3(FbblF]dl7&Fadl7$$!+r%*Q\"e'FbblFidl7$
Fidl$!+PdX;hFbblFddl7&Fhdl7$$!+T,V[_FbblF`el7$F`el$!+[:x5]FbblF[el7&F_
el7$$!+Hcu>RFbblFgel7$Fgel$!+R'R,#QFbblFbel7&Ffel7$$!+#HIIb#FbblF^fl7$
F^fl$!+tiQDDFbblFiel7&F]fl7$$!+!oE#\\8FbblFefl7$Fefl$!+Ao8X8FbblF`fl7&
Fdfl7$$\"+l!o5(f!#8F\\gl7$F\\gl$\"+5x1rfF^glFgfl7&F[gl7$$\"+)QLnO\"Fbb
lFdgl7$Fdgl$\"+aB[i8FbblF_gl7&Fcgl7$$\"+E+3yEFbblF[hl7$F[hl$\"+V@=YEFb
blFfgl7&Fjgl7$$\"+.%3*oQFbblFbhl7$Fbhl$\"+A*3Jx$FbblF]hl7&Fahl7$$\"+A&
=\\G&FbblFihl7$Fihl$\"+i_JU]FbblFdhl7&Fhhl7$$\"+K[Y%['FbblF`il7$F`il$
\"+k%*[RgFbblF[il7&F_il7$$\"+dB#)zyFbblFgil7$Fgil$\"+LjJ*3(FbblFbil7&F
fil7$$\"++-&\\6*FbblF^jl7$F^jl$\"+#>/U!zFbblFiil7&F]jl7$$\"+?\"3q/\"F[
blFejl7$Fejl$\"+GnIf')FbblF`jl7&Fdjl7$$\"+Q'[g<\"F[blF\\[m7$F\\[m$\"+j
i$4B*FbblFgjl7&F[[m7$$\"+!)))o58F[blFc[m7$Fc[m$\"+-Nij'*FbblF^[m7&Fb[m
7$$\"+`1LM9F[blFj[m7$Fj[m$\"+:+.2**FbblFe[m7&Fi[m7$$\"+Cjzq:F[blFa\\m7
$Fa\\mFa`lF\\\\m-Fg_l6&F[u$\"\"*Fd_lFf\\mFf\\m-%&STYLEG6#%,PATCHNOGRID
G-%+AXESLABELSG6%Q\"x6\"Q!F`]m-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$!$v\"!\"
#$\"$v\"F\\^mFe]m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve
 6" "Curve 7" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The graph
 of  " }{XPPEDIT 18 0 "y = sin*x;" "6#/%\"yG*&%$sinG\"\"\"%\"xGF'" }
{TEXT -1 13 " is drawn in " }{TEXT 256 4 "blue" }{TEXT -1 22 ", while \+
the graph of  " }{XPPEDIT 18 0 "y = x;" "6#/%\"yG%\"xG" }{TEXT -1 13 "
 is drawn in " }{TEXT 260 3 "red" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 23 "The picture shows that " }{XPPEDIT 18 0 "sin*x <= x;" "6#
1*&%$sinG\"\"\"%\"xGF&F'" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "0 <= x;
" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= Pi/2;" "6#1%!G*&%#PiG\"\"\"\"
\"#!\"\"" }{TEXT -1 79 ", so the area of the region enclosed between t
he curves over the interval from " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"
\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = Pi/2;" "6#/%\"xG*&%#PiG\"
\"\"\"\"#!\"\"" }{TEXT -1 5 " is: " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "Int(``(x-sin*x),x = 0 .. Pi/2) = x^2/2+cos*x;" "6#/-
%$IntG6$-%!G6#,&%\"xG\"\"\"*&%$sinGF,F+F,!\"\"/F+;\"\"!*&%#PiGF,\"\"#F
/,&*&F+F5F5F/F,*&%$cosGF,F+F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECE
WISE([Pi/2, ``],[0, ``]);" "6#-%*PIECEWISEG6$7$*&%#PiG\"\"\"\"\"#!\"\"
%!G7$\"\"!F," }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = ``(``(1/2)*``(Pi^2/4)+0)-(0+cos*0);" "6#/%!G,&-F$6
#,&*&-F$6#*&\"\"\"F-\"\"#!\"\"F--F$6#*&%#PiGF.\"\"%F/F-F-\"\"!F-F-,&F5
F-*&%$cosGF-F5F-F-F/" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "`` = Pi^2/8-1;" "6#/%!G,&*&%#PiG\"\"#\"\")!\"\"\"\"
\"F+F*" }{TEXT -1 1 " " }{TEXT 310 1 "~" }{TEXT -1 15 " 0.2337005501. \+
" }}{PARA 0 "" 0 "" {TEXT -1 110 "By the symmetry of the last picture \+
the area of the region enclosed between the curves over the interval f
rom " }{XPPEDIT 18 0 "x = -Pi/2;" "6#/%\"xG,$*&%#PiG\"\"\"\"\"#!\"\"F*
" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT 
-1 9 " is also " }{XPPEDIT 18 0 "Pi^2/8-1" "6#,&*&%#PiG\"\"#\"\")!\"\"
\"\"\"F)F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "x <= sin*x;" "6#1%\"xG*&
%$sinG\"\"\"F$F'" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "-Pi/2 <= x;" "6#
1,$*&%#PiG\"\"\"\"\"#!\"\"F)%\"xG" }{XPPEDIT 18 0 "`` <= 0;" "6#1%!G\"
\"!" }{TEXT -1 76 ", the area of the region enclosed between the curve
s over the interval from " }{XPPEDIT 18 0 "x = -Pi/2;" "6#/%\"xG,$*&%#
PiG\"\"\"\"\"#!\"\"F*" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 0;" "6#/
%\"xG\"\"!" }{TEXT -1 19 " is also given by: " }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(sin*x-x),x = -Pi/2 .. 0) = -cos*
x-x^2/2;" "6#/-%$IntG6$-%!G6#,&*&%$sinG\"\"\"%\"xGF-F-F.!\"\"/F.;,$*&%
#PiGF-\"\"#F/F/\"\"!,&*&%$cosGF-F.F-F/*&F.F5F5F/F/" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "PIECEWISE([0, ``],[-Pi/2, ``]);" "6#-%*PIECEWISEG6$7$\"
\"!%!G7$,$*&%#PiG\"\"\"\"\"#!\"\"F/F(" }{TEXT -1 1 " " }}{PARA 257 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(-cos*0-0)-(-cos(-Pi/2)-``(
1/2)*``(Pi^2/4));" "6#/%!G,&-F$6#,&*&%$cosG\"\"\"\"\"!F+!\"\"F,F-F+,&-
F*6#,$*&%#PiGF+\"\"#F-F-F-*&-F$6#*&F+F+F4F-F+-F$6#*&F3F4\"\"%F-F+F-F-
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(-1)+Pi^2/8;" "6#/%!G,&-F$6#,$\"
\"\"!\"\"F)*&%#PiG\"\"#\"\")F*F)" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi^2/8-1;" "6#/%!G,&*&%#PiG\"\"#\"
\")!\"\"\"\"\"F+F*" }{TEXT -1 1 " " }{TEXT 298 1 "~" }{TEXT -1 15 " 0.
2337005501. " }}{PARA 258 "" 0 "" {TEXT -1 61 "The total area of the r
egion enclosed between the graphs of  " }{XPPEDIT 18 0 "y = sin*x;" "6
#/%\"yG*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "y \+
= x;" "6#/%\"yG%\"xG" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "x = -Pi/2;
" "6#/%\"xG,$*&%#PiG\"\"\"\"\"#!\"\"F*" }{TEXT -1 4 " to " }{XPPEDIT 
18 0 "x = Pi/2;" "6#/%\"xG*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 14 " is t
herefore " }{XPPEDIT 18 0 "Pi^2/4-2" "6#,&*&%#PiG\"\"#\"\"%!\"\"\"\"\"
F&F(" }{TEXT -1 1 " " }{TEXT 311 1 "~" }{TEXT -1 16 "  0.4674011003. \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 79 "Int(sin(x)-x,x=-Pi/2..0)+Int(x-sin(x),x=0..Pi/2);\nvalue(%);\n
evalf(evalf(%,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6$,&-%
$sinG6#%\"xG\"\"\"F+!\"\"/F+;,$*&\"\"#F-%#PiGF,F-\"\"!F,-F%6$,&F+F,F(F
-/F+;F4,$*&F2F-F3F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"#!\"
\"*&\"\"%F%%#PiGF$\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+.5,uY!
#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "T
asks" }}{PARA 0 "" 0 "" {TEXT -1 90 "In questions 1 to 9 find the area
 of the given region. Illustrate your answer graphically." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 65 "Calculate the area of the re
gion enclosed between the graphs of  " }{XPPEDIT 18 0 "y = 9-x^2;" "6#
/%\"yG,&\"\"*\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 7 "  and  " }{XPPEDIT 
18 0 "y = 2-x;" "6#/%\"yG,&\"\"#\"\"\"%\"xG!\"\"" }{TEXT -1 6 " from \+
" }{XPPEDIT 18 0 "x = -2;" "6#/%\"xG,$\"\"#!\"\"" }{TEXT -1 4 " to " }
{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}
{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(7+x-x^2,x = -2 .. 2
) =68/3" "6#/-%$IntG6$,(\"\"(\"\"\"%\"xGF)*$F*\"\"#!\"\"/F*;,$F,F-F,*&
\"#oF)\"\"$F-" }{TEXT -1 2 "  " }}}{PARA 0 "" 0 "" {TEXT -1 46 "______
________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 2 "  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 46 "_______________________________
_______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 
65 "Calculate the area of the region enclosed between the graphs of  \+
" }{XPPEDIT 18 0 "y = 8+2*x-x^2;" "6#/%\"yG,(\"\")\"\"\"*&\"\"#F'%\"xG
F'F'*$F*F)!\"\"" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "y = x+2;" "6#/%
\"yG,&%\"xG\"\"\"\"\"#F'" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "x=-1" "
6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 3;" "6#/
%\"xG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 
1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 "  " 
}{XPPEDIT 18 0 "Int(6+x-x^2,x = -1 .. 3)=56/3" "6#/-%$IntG6$,(\"\"'\"
\"\"%\"xGF)*$F*\"\"#!\"\"/F*;,$F)F-\"\"$*&\"#cF)F1F-" }{TEXT -1 2 "  \+
" }}}{PARA 0 "" 0 "" {TEXT -1 46 "____________________________________
__________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 46 "______________________________________________" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 65 "Calculate the area o
f the region enclosed between the graphs of  " }{XPPEDIT 18 0 "y = exp
(-x);" "6#/%\"yG-%$expG6#,$%\"xG!\"\"" }{TEXT -1 7 "  and  " }
{XPPEDIT 18 0 "y = sqrt(x);" "6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT -1 6 " f
rom " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 4 " to " }
{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}
{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "Int(sqrt(x)-exp(-x),x \+
= 1 .. 2) = 4/3*sqrt(2)+exp(-2)-2/3-exp(-1);" "6#/-%$IntG6$,&-%%sqrtG6
#%\"xG\"\"\"-%$expG6#,$F+!\"\"F1/F+;F,\"\"#,**(\"\"%F,\"\"$F1-F)6#F4F,
F,-F.6#,$F4F1F,*&F4F,F8F1F1-F.6#,$F,F1F1" }{TEXT -1 1 " " }{TEXT 301 
1 "~" }{TEXT -1 10 "  0.98632 " }}}{PARA 0 "" 0 "" {TEXT -1 46 "______
________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 2 "  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 46 "_______________________________
_______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 
65 "Calculate the area of the region enclosed between the graphs of  \+
" }{XPPEDIT 18 0 "y = x^3+x;" "6#/%\"yG,&*$%\"xG\"\"$\"\"\"F'F)" }
{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "y = x^2+2*x+2;" "6#/%\"yG,(*$%\"x
G\"\"#\"\"\"*&F(F)F'F)F)F(F)" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "x =
 -1;" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x = 1
;" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hin
t: " }{XPPEDIT 18 0 "x^3+x<x^2+2*x+2" "6#2,&*$%\"xG\"\"$\"\"\"F&F(,(*$
F&\"\"#F(*&F+F(F&F(F(F+F(" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "-1<=x" 
"6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT 
-1 2 ". " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "
" {TEXT -1 2 "  " }{XPPEDIT 18 0 "Int(-x^3+x^2+x+2,x = -1 .. 1)=14/3" 
"6#/-%$IntG6$,**$%\"xG\"\"$!\"\"*$F)\"\"#\"\"\"F)F.F-F./F);,$F.F+F.*&
\"#9F.F*F+" }{TEXT -1 2 "  " }}}{PARA 0 "" 0 "" {TEXT -1 46 "_________
_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 2 "  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 46 "_______________________________
_______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 
56 "Find the area of the region enclosed between the curves " }
{XPPEDIT 18 0 "y = 6*x-x^2;" "6#/%\"yG,&*&\"\"'\"\"\"%\"xGF(F(*$F)\"\"
#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = x^2-4*x;" "6#/%\"yG,&*
$%\"xG\"\"#\"\"\"*&\"\"%F)F'F)!\"\"" }{TEXT -1 39 " between their poin
ts of intersection. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 
5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(10*x-2*x^2,x = 0 .. 5)=125/3" "6#/-%$IntG6$,&*&\"#5
\"\"\"%\"xGF*F**&\"\"#F**$F+F-F*!\"\"/F+;\"\"!\"\"&*&\"$D\"F*\"\"$F/" 
}{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 46 "______________________
________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 2 "  \+
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 46 "_________________________________________
_____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 57 "Find t
he area of the region enclosed between the curves  " }{XPPEDIT 18 0 "y
 = 2*x^2-3*x-8;" "6#/%\"yG,(*&\"\"#\"\"\"*$%\"xGF'F(F(*&\"\"$F(F*F(!\"
\"\"\")F-" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 2*x-x^2;" "6#/%\"yG
,&*&\"\"#\"\"\"%\"xGF(F(*$F)F'!\"\"" }{TEXT -1 39 " between their poin
ts of intersection. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 
5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(8+5*x-3*x^2,x = -1 .. 8/3)=1331/54" "6#/-%$IntG6$,(
\"\")\"\"\"*&\"\"&F)%\"xGF)F)*&\"\"$F)*$F,\"\"#F)!\"\"/F,;,$F)F1*&F(F)
F.F1*&\"%J8F)\"#aF1" }{TEXT -1 2 "  " }}}{PARA 0 "" 0 "" {TEXT -1 46 "
______________________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 2 "  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 46 "____________________
__________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 
";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" 
{TEXT -1 65 "Find the total area of the region(s) enclosed between the
 curves " }{XPPEDIT 18 0 "y = x^3-2*x+1;" "6#/%\"yG,(*$%\"xG\"\"$\"\"
\"*&\"\"#F)F'F)!\"\"F)F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = x^2+
1;" "6#/%\"yG,&*$%\"xG\"\"#\"\"\"F)F)" }{TEXT -1 37 " between any poin
ts of intersection. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 
5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(x^3-2*x-x^2,x = -1 .. 0)-Int(x^3-2*x-x^2,x = 0 .. 2
)=37/12" "6#/,&-%$IntG6$,(*$%\"xG\"\"$\"\"\"*&\"\"#F,F*F,!\"\"*$F*F.F/
/F*;,$F,F/\"\"!F,-F&6$,(*$F*F+F,*&F.F,F*F,F/*$F*F.F//F*;F4F.F/*&\"#PF,
\"#7F/" }{TEXT -1 3 "   " }}}{PARA 0 "" 0 "" {TEXT -1 46 "____________
__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 2 "  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 46 "_________________________________
_____________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q8 " }}{PARA 0 "" 0 "" {TEXT -1 
56 "Find the area of the region enclosed between the curves " }
{XPPEDIT 18 0 "y = x^4-8*x^2+16;" "6#/%\"yG,(*$%\"xG\"\"%\"\"\"*&\"\")
F)*$F'\"\"#F)!\"\"\"#;F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 16-x
^4;" "6#/%\"yG,&\"#;\"\"\"*$%\"xG\"\"%!\"\"" }{TEXT -1 37 " between an
y points of intersection. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 
{PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "Int(8*x^2-2*x^4,x = -2 .. 2)=256/15" "6#/-%$IntG6$,&*&
\"\")\"\"\"*$%\"xG\"\"#F*F**&F-F**$F,\"\"%F*!\"\"/F,;,$F-F1F-*&\"$c#F*
\"#:F1" }{TEXT -1 3 ".  " }}}{PARA 0 "" 0 "" {TEXT -1 46 "____________
__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 2 "  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 46 "_________________________________
_____________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q9 " }}{PARA 0 "" 0 "" {TEXT -1 
62 "Calculate the total area of the region bounded by the graphs  " }
{XPPEDIT 18 0 "y = 1/2;" "6#/%\"yG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 7 " \+
 and  " }{XPPEDIT 18 0 "y = cos(x);" "6#/%\"yG-%$cosG6#%\"xG" }{TEXT 
-1 78 " between their first two points of intersection which lie to th
e right of the " }{TEXT 299 1 "y" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 
0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "Int(1/2-cos(x),x=Pi/3..5*Pi/
3)=2/3*Pi+sqrt(3)" "6#/-%$IntG6$,&*&\"\"\"F)\"\"#!\"\"F)-%$cosG6#%\"xG
F+/F/;*&%#PiGF)\"\"$F+*(\"\"&F)F3F)F4F+,&*(F*F)F4F+F3F)F)-%%sqrtG6#F4F
)" }{TEXT -1 1 " " }{TEXT 300 1 "~" }{TEXT -1 11 " 3.82644   " }}}
{PARA 0 "" 0 "" {TEXT -1 46 "_________________________________________
_____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 46 "______________________________________________" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for \+
drawing pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{SECT 1 {PARA 0 "" 0 "" {TEXT -1 38 "Code for region between two graph
s - I" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1012 "g := x -> 2+sin(x/2)-cos(5/2*x)/4:\nf := x -> 5+cos
(x/3)-sin(3*x)/5:\na := .7: b := 3.4:\np1 := plot([f(x),g(x)],x=.5..3.
6,y=0..4,color=[red,blue],thickness=2):\np2 := plot([[[a,g(a)],[a,f(a)
]],[[b,g(b)],[b,f(b)]]],\n                color=COLOR(RGB,.4,.4,.4)):
\np3 := plot([[[a,0],[a,g(a)]],[[b,0],[b,g(b)]]],\n                col
or=COLOR(RGB,.5,.5,.5),linestyle=3):\np4 := plottools[arrow]([0,0],[4,
0],0,.1,.02,arrow,\n                color=black):\npp := plot(f(x),x=a
..b,adaptive=false,numpoints=25):\nfpts := op(1,op(1,pp)):\npp := plot
(g(x),x=a..b,adaptive=false,numpoints=25):\ngpts := op(1,op(1,pp)):\np
5 := plots[polygonplot]([seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2
..25)],\n           style=patchnogrid,color=COLOR(RGB,.85,.85,.85)):\n
t1 := plots[textplot]([2.4,6,`y = f(x)`],color=red):\nt2 := plots[text
plot]([2.4,2.45,`y = g(x)`],color=blue):\nt3 := plots[textplot]([[a,-.
2,`x=a`],[b,-.2,`x=b`],\n         [4.03,-.14,`x`]],color=black):\nplot
s[display]([p1,p2,p3,p4,p5,t1,t2,t3],view=[0..4.03,-.2..6.1],axes=none
);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 39 "Code for r
egion between two graphs - II" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 914 "g := x -> -2+sin(x/2)-cos(5
/2*x)/4:\nf := x -> 1+cos(x/3)-sin(3*x)/5:\na := .7: b := 3.4:\np1 := \+
plot([f(x),g(x)],x=.5..3.6,y=0..4,color=[red,blue],thickness=2):\np2 :
= plot([[[a,g(a)],[a,f(a)]],[[b,g(b)],[b,f(b)]]],\n                col
or=COLOR(RGB,.4,.4,.4)):\np3 := plot([[[a,-2.1],[a,g(a)]],[[b,-1.8],[b
,g(b)]]],\n                color=COLOR(RGB,.5,.5,.5),linestyle=3):\npp
 := plot(f(x),x=a..b,adaptive=false,numpoints=25):\nfpts := op(1,op(1,
pp)):\npp := plot(g(x),x=a..b,adaptive=false,numpoints=25):\ngpts := o
p(1,op(1,pp)):\np4 := plots[polygonplot]([seq([fpts[i-1],fpts[i],gpts[
i],gpts[i-1]],i=2..25)],\n           style=patchnogrid,color=COLOR(RGB
,.85,.85,.85)):\nt1 := plots[textplot]([2.4,2,`y = f(x)`],color=red):
\nt2 := plots[textplot]([2.4,-1.5,`y = g(x)`],color=blue):\nt3 := plot
s[textplot]([[a,-2.2,`x=a`],[b,-1.9,`x=b`]],color=black):\nplots[displ
ay]([p1,p2,p3,p4,t1,t2,t3],view=[-.2..4.03,-2.2..2.1],axes=none);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 42 "Code for sliced region \+
between two graphs " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 937 "g := x -> -2+sin(x/2)-cos(5/2*x)/4:\nf := \+
x -> 1+cos(x/3)-sin(3*x)/5:\na := .7: b := 3.4:\nn := 30: h := (b-a)/n
:\np1 := plot([f(x),g(x)],x=.5..3.6,y=0..4,color=[red,blue],thickness=
2):\np2 := plot([seq([[a+i*h,g(a+i*h)],[a+i*h,f(a+i*h)]],i=0..n)],\n  \+
      color=COLOR(RGB,.4,.4,.4)):\np3 := plot([[[a,-2.1],[a,g(a)]],[[b
,-1.8],[b,g(b)]]],\n                color=COLOR(RGB,.5,.5,.5),linestyl
e=3):\npp := plot(f(x),x=a..b,adaptive=false,numpoints=25):\nfpts := o
p(1,op(1,pp)):\npp := plot(g(x),x=a..b,adaptive=false,numpoints=25):\n
gpts := op(1,op(1,pp)):\np4 := plots[polygonplot]([seq([fpts[i-1],fpts
[i],gpts[i],gpts[i-1]],i=2..25)],\n           style=patchnogrid,color=
COLOR(RGB,.85,.85,.85)):\nt1 := plots[textplot]([2.4,2,`y = f(x)`],col
or=red):\nt2 := plots[textplot]([2.4,-1.5,`y = g(x)`],color=blue):\nt3
 := plots[textplot]([[a,-2.2,`x=a`],[b,-1.9,`x=b`]],color=black):\nplo
ts[display]([p1,p2,p3,p4,t1,t2,t3],view=[-.2..4.03,-2.2..2.1],axes=non
e);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 68 "Code for s
liced region between two graphs using genuine rectangles  " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1143 "g \+
:= x -> -2+sin(x/2)-cos(5/2*x)/4:\nf := x -> 1+cos(x/3)-sin(3*x)/5:\na
 := .7: b := 3.4:\nn := 30: h := (b-a)/n:\np1 := plot([f(x),g(x)],x=.5
..3.6,y=0..4,color=[red,blue],thickness=1):\nfvals := [seq(f(a+(2*i-1)
*h/2),i=1..n)]:\ngvals := [seq(g(a+(2*i-1)*h/2),i=1..n)]:\np2 := plot(
[[[a,-2.1],[a,gvals[1]]],[[b,-1.8],[b,gvals[n]]]],\n                co
lor=COLOR(RGB,.5,.5,.5),linestyle=3):\np3 := plot([[[a,gvals[1]],[a,fv
als[1]]],\n     seq([[a+i*h,min(gvals[i],gvals[i+1])],\n      [a+i*h,m
ax(fvals[i],fvals[i+1])]],i=1..n-1),\n      [[b,gvals[n]],[b,fvals[n]]
],\n     seq([[a+(i-1)*h,gvals[i]],[a+i*h,gvals[i]]],i=1..n),\n     se
q([[a+(i-1)*h,fvals[i]],[a+i*h,fvals[i]]],i=1..n)],\n        color=COL
OR(RGB,.4,.4,.4)):\np4 := plots[polygonplot]([seq([[a+(i-1)*h,fvals[i]
],[a+i*h,fvals[i]],\n    [a+i*h,gvals[i]],[a+(i-1)*h,gvals[i]]],i=1..n
)],\n           style=patchnogrid,color=COLOR(RGB,.85,.85,.85)):\nt1 :
= plots[textplot]([2.4,2,`y = f(x)`],color=red):\nt2 := plots[textplot
]([2.4,-1.5,`y = g(x)`],color=blue):\nt3 := plots[textplot]([[a,-2.2,`
x=a`],[b,-1.9,`x=b`]],color=black):\nplots[display]([p1,p2,p3,p4,t1,t2
,t3],view=[-.2..4.03,-2.2..2.1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 0 "" 0 "" {TEXT -1 52 "Code for area element for region between \+
two graphs " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1855 "g := x -> -2+sin(x/2)-cos(5/2*x)/4:\nf := x -> 1+
cos(x/3)-sin(3*x)/5:\na := .7: b := 3.4:\nc := 2: d := 2.1: e := (c+d)
/2: me := (f(e)+g(e))/2:\np1 := plot([f(x),g(x)],x=.2..4,y=0..4,color=
[red,blue],thickness=2):\np2 := plot([[[a,-2.1],[a,g(a)]],[[b,-1.8],[b
,g(b)]],\n   [[e,-1.6],[e,g(e)]]],color=black,linestyle=3):\np3 := plo
t([[[a,g(a)],[a,f(a)]],[[b,g(b)],[b,f(b)]]],\n           color=COLOR(R
GB,.4,.4,.4)):\npp := plot(f(x),x=c..d,adaptive=false,numpoints=5):\nf
pts := op(1,op(1,pp)):\npp := plot(g(x),x=c..d,adaptive=false,numpoint
s=5):\ngpts := op(1,op(1,pp)):\np4 := plots[polygonplot]([seq([fpts[i-
1],fpts[i],gpts[i],gpts[i-1]],i=2..5)],\n           style=patchnogrid,
color=COLOR(RGB,.6,.6,.6)):\npp := plot(f(x),x=a..b,adaptive=false,num
points=25):\nfpts := op(1,op(1,pp)):\npp := plot(g(x),x=a..b,adaptive=
false,numpoints=25):\ngpts := op(1,op(1,pp)):\np5 := plots[polygonplot
]([seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2..25)],\n           st
yle=patchnogrid,color=COLOR(RGB,.85,.85,.85)):\np6 := plot([[[c,g(c)],
[c,f(c)]],[[d,g(d)],[d,f(d)]]],color=black):\np7 := plot([[[0,g(e)],[e
,g(e)]],[[0,f(e)],[e,f(e)]]],\n                   color=black,linestyl
e=3):\np8 := plottools[arrow]([c-.2,me],[c,me],0,.1,.3,arrow,color=bla
ck):\np9 := plottools[arrow]([d+.2,me],[d,me],0,.1,.3,arrow,color=blac
k):\np10 := plottools[arrow]([.2,me+.15],[.2,f(e)],0,.07,.07,arrow,col
or=black):\np11 := plottools[arrow]([.2,me-.15],[.2,g(e)],0,.07,.07,ar
row,color=black):\nt1 := plots[textplot]([3.75,1.8,`y = f(x)`],color=r
ed):\nt2 := plots[textplot]([3.75,-1.05,`y = g(x)`],color=blue):\nt3 :
= plots[textplot]([[a,-2.2,`x=a`],[b,-1.9,`x=b`],\n          [.2,me,`f
(x) - g(x)`],[2.47,me,`x`],[e,-1.7,`x`]],color=black):\nt4 := plots[te
xtplot]([2.4,me,'D'],font=[SYMBOL,10]):\nplots[display]([p1,p2,p3,p4,p
5,p6,p7,p8,p9,p10,p11,\n               t1,t2,t3,t4],view=[-.2..4.03,-2
.2..2.1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }