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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 38 "Volumes of solids of revolution .
. III" }}{PARA 3 "" 0 "" {TEXT 281 58 " .. Volumes of revolution for a
 region between two curves " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter S
tone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 2
3.3.2007 " }}{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 308 100 "The volume of a solid of \+
revolution obtained when a region between two curves is rotated around
 the " }{TEXT 309 1 "x" }{TEXT 310 30 " axis\n .. the \"washer\" metho
d " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 40 "Suppose that the graphs of two function " }{XPPEDIT 18 0 
"f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x)" "6#
-%\"gG6#%\"xG" }{TEXT -1 15 " lie above the " }{TEXT 278 1 "x" }{TEXT 
-1 39 " axis, or at least do not go below the " }{TEXT 263 1 "x" }
{TEXT -1 29 " axis over the interval 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 suppose that the graph of " }{XPPEDIT 18 0 "f(x)" "
6#-%\"fG6#%\"xG" }{TEXT -1 25 " lies above the graph of " }{XPPEDIT 
18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 45 ", or at least does not go \+
below the graph of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT 
-1 31 ", throughout the interval 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 45 "Consider the region bounded b
y the graphs of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 
", " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 15 " and the li
nes " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 3 ",  " }
{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 60 "Then we can form a solid by rotating this region around
 the " }{TEXT 264 1 "x" }{TEXT -1 85 " axis through a complete turn. T
he solid of revolution will have a \"hole\" through it." }}{PARA 0 "" 
0 "" {TEXT -1 62 "As in the case of a single graph, subdivide the inte
rval from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to " }
{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 52 " into a large number
 of subintervals of equal width " }{XPPEDIT 18 0 "delta" "6#%&deltaG" 
}{TEXT 266 1 "x" }{TEXT -1 78 ", and consider the corresponding subdiv
ision of the region between 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 31 " into vertical strips of width " }
{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 265 1 "x" }{TEXT -1 2 ". " }
}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 484 391 391 {PLOTDATA 2 "
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47%7$$\"++++]TF4$\"+\"G`lE\"F4F[^t7$$\"++++]UF4Fb^tF;Fjo-F$6&7$7$Fg]t$
\"+b;!4#RF\\[s7$Fg]tF+7%7$Fe^t$\"+'\\qi<\"F\\[sF]_t7$F`^tF`_tF;Fjo-%%T
EXTG6%7$$\"#_F_p$FIF*Q\"b6\"F?-Fd_t6%7$F]p$FLF*Q\"aF[`tF?-Fd_t6%7$$\"#
bF_p$\"$&RF*Q)y~=~f(x)F[`tFicl-Fd_t6%7$$\"#eF_p$\"#FF_pQ)y~=~g(x)F[`tF
i]m-Fd_t6%7$$\"#HF_p$\"$X$F*Q\"xF[`tFjo-Fd_t6%7$$\"#LF_p$!#GF*FhatFjo-
Fd_t6%7$Fc[t$\"+5lOd:F4Q%f(x)F[`tFjo-Fd_t6%7$Fg]t$\"+b;!4#pF\\[sQ%g(x)
F[`tFjo-Fd_t6&7$$\"#GF_pFfatQ\"dF[`tFjo-%%FONTG6$%'SYMBOLG\"#5-%*AXESS
TYLEG6#%%NONEG-%+AXESLABELSG6%Q!F[`tF^dt-Fcct6#%(DEFAULTG-%%VIEWG6$;F(
FeclFadt" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Cur
ve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Cur
ve 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" "C
urve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve
 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33
" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "C
urve 40" "Curve 41" }}{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 37 "
Rotating a typical strip situated at " }{TEXT 268 1 "x" }{TEXT -1 12 "
 around the " }{TEXT 267 1 "x" }{TEXT -1 58 " axis generates a \"disc \+
with a hole\" or \"washer\" of width " }{XPPEDIT 18 0 "delta" "6#%&del
taG" }{TEXT 269 1 "x" }{TEXT -1 8 ".  This " }{TEXT 259 14 "volume ele
ment" }{TEXT -1 49 " may be regarded as being a cylinder with radius \+
" }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 35 " with a cylind
rical hole of radius " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT 
-1 26 ". Its volume is therefore:" }}{PARA 256 "" 0 "" {TEXT -1 2 "  \+
" }{XPPEDIT 18 0 "Pi*f(x)^2*`.`*delta" "6#**%#PiG\"\"\"*$-%\"fG6#%\"xG
\"\"#F%%\".GF%%&deltaGF%" }{TEXT 279 1 "x" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "- Pi*f(x)^2*`.`*delta" "6#,$**%#PiG\"\"\"*$-%\"fG6#%\"xG\"\"#F&%
\".GF&%&deltaGF&!\"\"" }{XPPEDIT 18 0 "x=Pi*(f(x)^2-g(x)^2)*`.`*delta
" "6#/%\"xG**%#PiG\"\"\",&*$-%\"fG6#F$\"\"#F'*$-%\"gG6#F$F-!\"\"F'%\".
GF'%&deltaGF'" }{TEXT 280 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "An estimate of the total
 volume of the solid of revolution can be obtained by adding the volum
es of such volume elements as " }{TEXT 273 1 "x" }{TEXT -1 13 " varies
 from " }{TEXT 274 1 "a" }{TEXT -1 4 " to " }{TEXT 275 1 "b" }{TEXT 
-1 64 ", that is, the volume can be approximated by a sum of the form:
 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(Pi*[f(x)^2-g
(x)^2]*delta,x = a .. b);" "6#-%$SumG6$*(%#PiG\"\"\"7#,&*$-%\"fG6#%\"x
G\"\"#F(*$-%\"gG6#F/F0!\"\"F(%&deltaGF(/F/;%\"aG%\"bG" }{TEXT 270 1 "x
" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 271 
1 "x" }{TEXT -1 94 " runs over a finite sequence of values giving the \+
locations of vertical strips of equal width " }{XPPEDIT 18 0 "delta" "
6#%&deltaG" }{TEXT 272 1 "x" }{TEXT -1 61 " which cover the whole of t
he region between the graphs from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"a
G" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 
2 ". " }}{PARA 0 "" 0 "" {TEXT -1 81 "Now consider taking progressivel
y narrower strips, and corresponding washers. As " }{XPPEDIT 18 0 "del
ta" "6#%&deltaG" }{XPPEDIT 18 0 "x->0" "6#f*6#%\"xG7\"6$%)operatorG%&a
rrowG6\"\"\"!F*F*F*" }{TEXT -1 83 " and the number of strips tends to \+
infinity, the sum above approaches the integral " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(Pi*[f(x)^2-g(x)^2],x = a .. b);" "6
#-%$IntG6$*&%#PiG\"\"\"7#,&*$-%\"fG6#%\"xG\"\"#F(*$-%\"gG6#F/F0!\"\"F(
/F/;%\"aG%\"bG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 87 "The vol
ume of the solid of revolution formed by rotating the region between t
he 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 12 " around the " }
{TEXT 276 1 "x" }{TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "V[x] = Pi*Int([f(x)^2-g(x)^2],x = a .. b);" "
6#/&%\"VG6#%\"xG*&%#PiG\"\"\"-%$IntG6$7#,&*$-%\"fG6#F'\"\"#F**$-%\"gG6
#F'F4!\"\"/F';%\"aG%\"bGF*" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{TEXT 277 15 "_______________" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }
{TEXT -1 38 ": One could compute this integral as  " }{XPPEDIT 18 0 "P
i*Int(f(x)^2,x = a .. b)-Pi*Int(g(x)^2,x = a .. b);" "6#,&*&%#PiG\"\"
\"-%$IntG6$*$-%\"fG6#%\"xG\"\"#/F.;%\"aG%\"bGF&F&*&F%F&-F(6$*$-%\"gG6#
F.F//F.;F2F3F&!\"\"" }{TEXT -1 68 ", which amounts to subtracting the \+
volume of region under the graph " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%
\"gG6#%\"xG" }{TEXT -1 53 " from the volume of the solid formed by reg
ion under " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 
2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 333 100 "The volu
me of a solid of revolution obtained when a region between two curves \+
is rotated around the " }{TEXT 334 1 "y" }{TEXT 335 40 " axis\n .. the
 \"cylindrical shell\" method" }{TEXT -1 2 "  " }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "Suppose that t
he graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " l
ies above the graph of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }
{TEXT -1 45 ", or at least does not go below the graph of " }{XPPEDIT 
18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 31 ", throughout the interval \+
from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to " }
{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 8 ", where " }{TEXT 323 
1 "a" }{TEXT -1 35 " is greater than or equal to 0 and " }{XPPEDIT 18 
0 "a<b" "6#2%\"aG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
45 "Consider the region bounded by the graphs of " }{XPPEDIT 18 0 "f(x
)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6
#%\"xG" }{TEXT -1 15 " and the lines " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%
\"aG" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 95 "Then we can form a solid by r
otating this region around the y axis through a complete turn. If " }
{XPPEDIT 18 0 "a<>0" "6#0%\"aG\"\"!" }{TEXT -1 56 ", the solid of revo
lution will have a \"hole\" through it." }}{PARA 0 "" 0 "" {TEXT -1 
28 "Subdivide the interval from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG
" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 
52 " into a large number of subintervals of equal width " }{XPPEDIT 
18 0 "delta" "6#%&deltaG" }{TEXT 312 1 "x" }{TEXT -1 78 ", and conside
r the corresponding subdivision of the region between 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 31 " into ver
tical strips of width " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 
311 1 "x" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
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3CFSJWg\\f>F*7$Faeq$\"3$Rt^\"4:.C>F*7$Ffeq$\"3sW/\"RH$)p)=F*7$F[fq$\"3
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[2;=F*7$F[[r$\"3Q0E#3>M2&=F*7$F`[r$\"3C%y)pU4?%)=F*7$Fe[r$\"3B[(4[[W=#
>F*7$Fj[r$\"3#3f:3)4<f>F*7$F^in$\"3+ChC++++?F*F[[l-Ffhp6#Fcbm-F$6%7$Fb
]q7$F^in$\"3!**************R#F*F[[lFehp-F$6%7$7$FgfoFiz7$FgfoFh[lF[[lF
ehp-F$6%7$Fg[l7$FgzFjerF[[lFehp-F$6%7$7$$\"3/++++++]5F*$\"3/+++++++=F*
7$Fifr$\"3;+++++++AF*F[[lFehp-F$6&7$7$$\"\"*Fa[lF_ao7$$\"#5Fa[lF_ao7%7
$$\"+++++(*Fh[m$\"++++IBF]]mFfgr7$F[hr$\"++++qAF]]mFdbmF[[l-F$6&7$7$$
\"#7Fa[lF_ao7$Fe_mF_ao7%7$$\"++++I6F]]mF`hrFhhr7$F[irF]hrFdbmF[[l-%%TE
XTG6&7$$\"#8Fa[lF^inQ,f(x)~-~g(x)6\"F[[l-%%FONTG6$%*HELVETICAGFhgr-F_i
r6&7$$\"$2\"F`bm$\"#DFa[lQ\"xFeirF[[lFfir-F_ir6&7$$\"+++++]Fh[mFh[lFaj
rF[[lFfir-F_ir6&7$$\"$.\"F`bmF_jrQ\"dFeirF[[l-Fgir6$Fi\\qFhgr-%+AXESLA
BELSG6%Q!FeirFb[s-Fgir6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$Fe[s
Fe[s" 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" "Cu
rve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve \+
21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" 
"Curve 28" "Curve 29" }}{TEXT -1 3 "   " }}{PARA 256 "" 0 "" {TEXT -1 
2 "  " }}{PARA 0 "" 0 "" {TEXT -1 5 "This " }{TEXT 259 14 "volume elem
ent" }{TEXT -1 72 " may be thought of as the \"thickened\" surface of \+
a cylinder with radius " }{TEXT 325 1 "x" }{TEXT -1 13 " and height  \+
" }{XPPEDIT 18 0 "f(x)-g(x);" "6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F'!\"\"
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 131 "Imagine that this cy
lindrical shell is sliced open with a vertical cut, and opened out to \+
form a (thickened) rectangle with length " }{XPPEDIT 18 0 "2*Pi*x;" "6
#*(\"\"#\"\"\"%#PiGF%%\"xGF%" }{TEXT -1 9 ", height " }{XPPEDIT 18 0 "
f(x)-g(x)" "6#,&-%\"fG6#%\"xG\"\"\"-%\"gG6#F'!\"\"" }{TEXT -1 15 " and
 thickness " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT 326 1 "x" }
{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 793 230 
230 {PLOTDATA 2 "6<-%'CURVESG6$7'7$$\"\"!F)F(7$F($\"\"#F)7$$\"3;++++++
+s!#<F+7$F.F(F'-%&COLORG6&%$RGBG$\"\"&!\"\"F)$\"\"(F8-F$6$7&F*7$$\"3o+
++++++q!#>$\"3%)************p?F07$$\"3e************psF0FBF-F2-F$6$7%FD
7$FEF?F1F2-%)POLYGONSG6%7&F'F*7$$\"#sF8F+7$FPF(-F36&F5$\"#&*!\"#$\"#\"
*FW\"\"\"-%&STYLEG6#%,PATCHNOGRIDG-FL6%7&F*7$$F:FW$\"$2#FW7$$\"$F(FWF^
oFO-F36&F5$\"$`*!\"$$\"$B*FgoFZFen-FL6%7&FOF`o7$FaoF]oFR-F36&F5FU$\"#*
)FWFZFen-F$6%7$F'7$F($!3))**************H!#=F2-%*LINESTYLEG6#F,-F$6%7$
F17$F.FfpF2Fip-F$6%7$F'7$$!3a**************pFhpF(F2Fip-F$6%7$F*7$FdqF+
F2Fip-F$6%7$F-7$$\"3M+++++++zF0F+F2Fip-F$6%7$FD7$$\"3k*************4)F
0FBF2Fip-F$6&7$7$$\"+++++L!\"*$FWF87$F(F]s7%7$$\")+++**F\\s$!+++++C!#5
F^s7$Fas$!+++++;FesFenF2-F$6&7$7$$\"+++++RF\\sF]s7$$\"+++++sF\\sF]s7%7
$$\"++++,rF\\sFgsF_t7$FdtFcsFenF2-F$6&7$7$$!\"&F8$\"$8\"FW7$F[u$\"$+#F
W7%7$$!+++++bFes$\"+++5R>F\\sF_u7$$!+++++XFesFfuFenF2-F$6&7$7$F[u$\"#(
)FW7$F[uF(7%7$Fiu$\"*+++4'FesFav7$FduFdvFenF2-F$6&7$7$$\"#vF8$\"#=F87$
$\"#xF8$\"#?F87%7$$\"+j7D:wF\\s$\"+P([Z'>F\\sF_w7$$\"+P([Zm(F\\s$\"+j7
D:>F\\sFenF2-F$6&7$7$$\"$(zFW$\"$F#FW7$$\"$x(FWF^o7%7$$\"+P([Z&yF\\s$
\"+j7D0@F\\sFgx7$$\"+j7D0yF\\s$\"+P([Z:#F\\sFenF2-%%TEXTG6&7$$\"+++++O
F\\sF]sQ'2~~~~x6\"F2-%%FONTG6$%*HELVETICAG\"#5-Ffy6&7$F`w$\"#AF8Q\"xF
\\zF2F]z-Ffy6&7$F[u$FZF)Q,f(x)~-~g(x)F\\zF2F]z-Ffy6&7$$\"++++5OF\\sF]s
Q\"pF\\zF2-F^z6$%'SYMBOLGFaz-Ffy6&7$$\"$i(FWFezQ\"dF\\zF2Fc[l-%+AXESLA
BELSG6%Q!F\\zF_\\l-F^z6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$Fb\\
lFb\\l" 1 2 0 1 10 0 2 9 1 1 2 1.000000 46.000000 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" "Curv
e 21" "Curve 22" "Curve 23" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 75 "This indicates that we should take the volume of the volume ele
ment to be: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*Pi*
x*(f(x)-g(x))*delta" "6#*,\"\"#\"\"\"%#PiGF%%\"xGF%,&-%\"fG6#F'F%-%\"g
G6#F'!\"\"F%%&deltaGF%" }{TEXT 327 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Another way to see
 this is as follows. If " }{TEXT 330 1 "x" }{TEXT -1 26 " is the dista
nce from the " }{TEXT 331 1 "y" }{TEXT -1 58 " axis to the mid-point o
f the subinterval of the interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%
\"bG" }{TEXT -1 141 " which determines the cylindrical shell, the volu
me of the cylindrical shell is obtained by subtracting the volume of a
 cylinder with height " }{XPPEDIT 18 0 "h(x) = f(x)-g(x);" "6#/-%\"hG6
#%\"xG,&-%\"fG6#F'\"\"\"-%\"gG6#F'!\"\"" }{TEXT -1 12 " and radius " }
{XPPEDIT 18 0 "x-delta*x/2" "6#,&%\"xG\"\"\"*(%&deltaGF%F$F%\"\"#!\"\"
F)" }{TEXT -1 43 " from the volume of a cylinder with height " }
{XPPEDIT 18 0 "h(x) = f(x)-g(x);" "6#/-%\"hG6#%\"xG,&-%\"fG6#F'\"\"\"-
%\"gG6#F'!\"\"" }{TEXT -1 12 " and radius " }{XPPEDIT 18 0 "x+delta*x/
2" "6#,&%\"xG\"\"\"*(%&deltaGF%F$F%\"\"#!\"\"F%" }{TEXT -1 14 ". This \+
gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Pi*(x+delt
a*x/2)^2*h(x)-Pi*(x-delta*x/2)^2*h(x);" "6#,&*(%#PiG\"\"\"*$,&%\"xGF&*
(%&deltaGF&F)F&\"\"#!\"\"F&F,F&-%\"hG6#F)F&F&*(F%F&*$,&F)F&*(F+F&F)F&F
,F-F-F,F&-F/6#F)F&F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "`` = Pi*((x+delta*x/2)^2-(x+delta*x/2)^2)*h(x);" "6
#/%!G*(%#PiG\"\"\",&*$,&%\"xGF'*(%&deltaGF'F+F'\"\"#!\"\"F'F.F'*$,&F+F
'*(F-F'F+F'F.F/F'F.F/F'-%\"hG6#F+F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 
"" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*((x^2+x*delta*x+delta*x^2/4)
^2-(x^2-x*delta*x+delta*x^2/4))*h(x);" "6#/%!G*(%#PiG\"\"\",&*$,(*$%\"
xG\"\"#F'*(F,F'%&deltaGF'F,F'F'*(F/F'*$F,F-F'\"\"%!\"\"F'F-F',(*$F,F-F
'*(F,F'F/F'F,F'F3*(F/F'*$F,F-F'F2F3F'F3F'-%\"hG6#F,F'" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = Pi*``(2*x*delta*x)*h(x);" "6#/%!G*(%#PiG\"\"\"-F$6
#**\"\"#F'%\"xGF'%&deltaGF'F,F'F'-%\"hG6#F,F'" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 2*Pi*x*(f(x)-g(x))*delta;" "6#/%!G*,\"\"#\"\"\"%#P
iGF'%\"xGF',&-%\"fG6#F)F'-%\"gG6#F)!\"\"F'%&deltaGF'" }{TEXT 332 1 "x
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 124 "An estimate of the total volume of the solid of revoluti
on can be obtained by adding the volumes of such volume elements as " 
}{TEXT 318 1 "x" }{TEXT -1 13 " varies from " }{TEXT 319 1 "a" }{TEXT 
-1 4 " to " }{TEXT 320 1 "b" }{TEXT -1 64 ", that is, the volume can b
e approximated by a sum of the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "Sum(2*Pi*x*(f(x)-g(x))*delta,x = a .. b);" "6#-%$S
umG6$*,\"\"#\"\"\"%#PiGF(%\"xGF(,&-%\"fG6#F*F(-%\"gG6#F*!\"\"F(%&delta
GF(/F*;%\"aG%\"bG" }{TEXT 315 1 "x" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "
" {TEXT -1 6 "where " }{TEXT 316 1 "x" }{TEXT -1 94 " runs over a fini
te sequence of values giving the locations of vertical strips of equal
 width " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT 317 1 "x" }{TEXT 
-1 61 " which cover the whole of the region between the 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 
102 "Now consider taking progressively narrower strips, and correspond
ingly thinner cylindrical shells. As " }{XPPEDIT 18 0 "delta" "6#%&del
taG" }{XPPEDIT 18 0 "x->0" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"
!F*F*F*" }{TEXT -1 83 " and the number of strips tends to infinity, th
e sum above approaches the integral " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "Int(2*Pi*x*(f(x)-g(x)),x = a .. b);" "6#-%$IntG6$**
\"\"#\"\"\"%#PiGF(%\"xGF(,&-%\"fG6#F*F(-%\"gG6#F*!\"\"F(/F*;%\"aG%\"bG
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 87 "The volume of the sol
id of revolution formed by rotating the region between 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 12 " around the " }{TEXT 321 1 "y" }
{TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "V[y] = 2*Pi*Int(x*(f(x)-g(x)),x = a .. b);" "6#/&%\"VG6#%\"yG*(
\"\"#\"\"\"%#PiGF*-%$IntG6$*&%\"xGF*,&-%\"fG6#F0F*-%\"gG6#F0!\"\"F*/F0
;%\"aG%\"bGF*" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{TEXT 322 17 "_________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "In the special case that \+
" }{XPPEDIT 18 0 "g(x)=0" "6#/-%\"gG6#%\"xG\"\"!" }{TEXT -1 62 ", the \+
region to be rotated is just the region under the graph " }{XPPEDIT 
18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"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 96 ". In this case the volume of the solid of r
evolution obtained by rotating the region around the " }{TEXT 329 1 "y
" }{TEXT -1 10 " axis is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "V[y] = 2*Pi*Int(x*f(x),x = a .. b);" "6#/&%\"VG6#%\"yG*
(\"\"#\"\"\"%#PiGF*-%$IntG6$*&%\"xGF*-%\"fG6#F0F*/F0;%\"aG%\"bGF*" }
{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 328 11 "____
_______" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }
}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1a" }}{PARA 0 "" 0 "" 
{TEXT 283 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 89 
"Find the volume of the solid of revolution formed when the region bou
nded by the curves  " }{XPPEDIT 18 0 "y = sqrt(x);" "6#/%\"yG-%%sqrtG6
#%\"xG" }{TEXT -1 6 "  and " }{XPPEDIT 18 0 "y=1/x" "6#/%\"yG*&\"\"\"F
&%\"xG!\"\"" }{TEXT -1 16 ", and the lines " }{XPPEDIT 18 0 "x = 2;" "
6#/%\"xG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 4;" "6#/%\"xG\"
\"%" }{TEXT -1 23 " is rotated around the " }{TEXT 282 1 "x" }{TEXT 
-1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT 284 8 "Solution" }{TEXT -1 2 ":
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 917 "f := x -> sqrt(x):\ng := x -> 1/x:\np1 := plot([f(x)
,-f(x),g(x),-g(x)],x=0..5,y=-2.3..2.3,\n   color=[red$2,blue$2],thickn
ess=[2,1,2,1],linestyle=[1,2,1,2]):\na := 2: b := 4:\npp := plot(f(x),
x=a..b,adaptive=false,numpoints=25):\nfpts := op(1,op(1,pp)):\npp := p
lot(g(x),x=a..b,adaptive=false,numpoints=25):\ngpts := op(1,op(1,pp)):
\np2 := plots[polygonplot]([seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],
i=2..25)],\n           style=patchnogrid,color=COLOR(RGB,.95,.95,.95))
:\np3 := plot([[[a,g(a)],[a,f(a)]],[[b,g(b)],[b,f(b)]],\n      [[a,0],
[a,g(a)]],[[b,0],[b,g(b)]]],linestyle=[1$2,2$2],color=black):\nh := ev
alf(Pi/20): r := .2:\np4 := plot([[seq([a+r*f(a)*cos(i*h),f(a)*sin(i*h
)],i=0..40)],\n       [seq([a+r*g(a)*cos(i*h),g(a)*sin(i*h)],i=0..40)]
,\n       [seq([b+r*f(b)*cos(i*h),f(b)*sin(i*h)],i=0..40)],\n       [s
eq([b+r*g(b)*cos(i*h),g(b)*sin(i*h)],i=0..40)]],\n       linestyle=2,c
olor=black):\nplots[display]([p1,p2,p3,p4]);" }}{PARA 13 "" 1 "" 
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\\hr$!3)******\\&pwn<F07$Fggr$!3#******f[UD-#F07$Fbgr$!39+++6j^FAF07$F
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****4&3ApCF07$F`fr$!3#********GTwP#F07$F[fr$!31+++5j^FAF07$Ffer$!3.+++
%[UD-#F07$Faer$!3++++_pwn<F07$F\\er$!3%*******HJYp9F07$Fgdr$!3-+++[i(
\\8\"F07$Fbdr$!3k*****>$[UDxF-7$F]dr$!3=+++7;'3\"RF-7$Fjcr$\"33+++3Q.^
?F_jpFibpFjfl-%+AXESLABELSG6%Q\"x6\"Q\"yFg]s-%%FONTG6#%(DEFAULTG-%%VIE
WG6$;F(Fg[l;$!#B!\"\"$\"#BFd^s" 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" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur
ve 12" "Curve 13" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }
{XPPEDIT 18 0 "f(x) = sqrt(x);" "6#/-%\"fG6#%\"xG-%%sqrtG6#F'" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "g(x)=1/x" "6#/-%\"gG6#%\"xG*&\"\"\"F)F'!
\"\"" }{TEXT -1 32 ".  Then the required volume is: " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V[x] = Pi*Int([f(x)^2-g(x)^2],x = \+
2 .. 4);" "6#/&%\"VG6#%\"xG*&%#PiG\"\"\"-%$IntG6$7#,&*$-%\"fG6#F'\"\"#
F**$-%\"gG6#F'F4!\"\"/F';F4\"\"%F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*Int(``(sqrt(x)^2-(1/x)^2),x =
 2 .. 4);" "6#/%!G*&%#PiG\"\"\"-%$IntG6$-F$6#,&*$-%%sqrtG6#%\"xG\"\"#F
'*$*&F'F'F2!\"\"F3F6/F2;F3\"\"%F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*Int(``(x-1/(x^2)),x = 2 .. 4)
;" "6#/%!G*&%#PiG\"\"\"-%$IntG6$-F$6#,&%\"xGF'*&F'F'*$F.\"\"#!\"\"F2/F
.;F1\"\"%F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = Pi*(x^2/2+1/x);" "6#/%!G*&%#PiG\"\"\",&*&%\"xG\"\"
#F+!\"\"F'*&F'F'F*F,F'F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``;" "6#%!G
" }{XPPEDIT 18 0 "PIECEWISE([4, ``],[``, ``],[2, ``]);" "6#-%*PIECEWIS
EG6%7$\"\"%%!G7$F(F(7$\"\"#F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+
Pi*(``(8+1/4)-(2+1/2));" "6#/%!G*&%#PiG\"\"\",&-F$6#,&\"\")F'*&F'F'\"
\"%!\"\"F'F',&\"\"#F'*&F'F'F1F/F'F/F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``
= Pi*(6-1/4)" "6#/%!G*&%#PiG\"\"\",&\"\"'F'*&F'F'\"\"%!\"\"F,F'" }
{XPPEDIT 18 0 "``=23*Pi/4" "6#/%!G*(\"#B\"\"\"%#PiGF'\"\"%!\"\"" }
{TEXT -1 1 " " }{TEXT 285 1 "~" }{TEXT -1 14 " 18.06415776. " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "f \+
:= x -> sqrt(x);\ng := x -> 1/x;\nPi*Int(f(x)^2-g(x)^2,x=2..4);\nvalue
(%);\nevalf(evalf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%%s
qrtG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operator
G%&arrowGF(*&\"\"\"F-9$!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
*&%#PiG\"\"\"-%$IntG6$,&%\"xGF%*&F%F%*$)F*\"\"#F%!\"\"F//F*;F.\"\"%F%
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#B\"\"\"\"\"%!\"\"%#PiGF&F&
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+wdT1=!\")" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 1b " }}{PARA 0 "" 0 "" {TEXT 
336 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 89 "Find \+
the volume of the solid of revolution formed when the region bounded b
y the curves  " }{XPPEDIT 18 0 "y = sqrt(x);" "6#/%\"yG-%%sqrtG6#%\"xG
" }{TEXT -1 6 "  and " }{XPPEDIT 18 0 "y=1/x" "6#/%\"yG*&\"\"\"F&%\"xG
!\"\"" }{TEXT -1 16 ", and the lines " }{XPPEDIT 18 0 "x = 2;" "6#/%\"
xG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 4;" "6#/%\"xG\"\"%" }
{TEXT -1 23 " is rotated around the " }{TEXT 339 1 "y" }{TEXT -1 8 " a
xis.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 337 8 
"Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1570 "f := x -> sqrt(x):\ng := x -> 1/x
:\np1 := plot([f(x),g(x)],x=0..5,y=-.3..2.6,\n   color=[red,blue],thic
kness=2,linestyle=1):\np2 := plot([f(-x),g(-x)],x=-5..0,y=-.3..2.6,\n \+
  color=[red,blue],thickness=1,linestyle=2):\na := 2: b := 4: c := 3.3
:\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)):\np3 := plots[polygonplot]([seq([fpts[i-1],fpts[i],
gpts[i],gpts[i-1]],i=2..25)],\n           style=patchnogrid,color=COLO
R(RGB,.9,.9,.9)):\npp := plot(f(-x),x=-b..-a,adaptive=false,numpoints=
25):\nfpts := op(1,op(1,pp)):\npp := plot(g(-x),x=-b..-a,adaptive=fals
e,numpoints=25):\ngpts := op(1,op(1,pp)):\np4 := plots[polygonplot]([s
eq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2..25)],\n           style=
patchnogrid,color=COLOR(RGB,.92,.92,.92)):\np5 := plot([[[a,g(a)],[a,f
(a)]],[[b,g(b)],[b,f(b)]],\n      [[a,0],[a,g(a)]],[[b,0],[b,g(b)]],\n
    [[-a,0],[-a,f(a)]],[[-b,0],[-b,f(b)]]],linestyle=[1$2,2$4],color=b
lack):\nh := evalf(Pi/20): r := .12:\np6 := plot([[seq([a*cos(i*h),f(a
)+r*a*sin(i*h)],i=0..40)],\n       [seq([a*cos(i*h),g(a)+r*a*sin(i*h)]
,i=0..40)],\n       [seq([b*cos(i*h),f(b)+r*b*sin(i*h)],i=0..40)],\n  \+
     [seq([b*cos(i*h),g(b)+r*b*sin(i*h)],i=0..40)]],\n       linestyle
=2,color=black):\np7 := plot([[seq([c*cos(i*h),f(c)+r*c*sin(i*h)],i=0.
.40)],\n           [seq([c*cos(i*h),g(c)+r*c*sin(i*h)],i=0..40)],\n   \+
        [[c,g(c)],[c,f(c)]],[[-c,g(c)],[-c,f(c)]]],\n       linestyle=
1,color=COLOR(RGB,.7,.7,.7),thickness=2): \nplots[display]([p1,p2,p3,p
4,p5,p6,p7],tickmarks=[5,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 758 223 
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R(*Rc\"F0$\"3'[^6woSZ&RF07$$\"3umm\"H2P\"Q?F0$\"3O(3t-:tX^%F07$$\"3MLL
$eRwX5$F0$\"3ITb(>ds=d&F07$$\"33ML$3x%3yTF0$\"3%4EH#=1\"QY'F07$$\"3emm
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***\\il'pisF0$\"3`Z7&[lX@_)F07$$\"3>MLe*)>VB$)F0$\"3H6xc!\\%GB\"*F07$$
\"3Y++DJbw!Q*F0$\"3qf2W:_V&o*F07$$\"3%ommTIOo/\"!#<$\"3i#[:l=]J-\"Fjo7
$$\"3YLL3_>jU6Fjo$\"3iD17R'R*o5Fjo7$$\"37++]i^Z]7Fjo$\"3L![*onkC=6Fjo7
$$\"33++](=h(e8Fjo$\"37$\\%[@\"fc;\"Fjo7$$\"3/++]P[6j9Fjo$\"3UsQbRGf47
Fjo7$$\"3UL$e*[z(yb\"Fjo$\"3ymDI5)\\\"[7Fjo7$$\"3wmm;a/cq;Fjo$\"3_g!pj
i,DH\"Fjo7$$\"3%ommmJ<gw\"Fjo$\"3`mCeUe\"*G8Fjo7$$\"3/+]iSj0x=Fjo$\"3a
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[l;$!\"$F_cq$\"#EF_cq" 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" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu
rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+
19" "Curve 20" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }
{XPPEDIT 18 0 "f(x) = sqrt(x);" "6#/-%\"fG6#%\"xG-%%sqrtG6#F'" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "g(x)=1/x" "6#/-%\"gG6#%\"xG*&\"\"\"F)F'!
\"\"" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 49 "Using cylindric
al shells the required volume is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "V[y] = 2*Pi*Int(x*(f(x)-g(x)),x = 2 .. 4);" "6#/&%\"
VG6#%\"yG*(\"\"#\"\"\"%#PiGF*-%$IntG6$*&%\"xGF*,&-%\"fG6#F0F*-%\"gG6#F
0!\"\"F*/F0;F)\"\"%F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "`` = 2*Pi*Int(x*(sqrt(x)-1/x),x = 2 .. 4);" "6#/%!
G*(\"\"#\"\"\"%#PiGF'-%$IntG6$*&%\"xGF',&-%%sqrtG6#F-F'*&F'F'F-!\"\"F3
F'/F-;F&\"\"%F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 2*Pi*Int(``(x*sqrt(x)-1),x = 2 .. 4);" "6#/%!G*(\"
\"#\"\"\"%#PiGF'-%$IntG6$-F$6#,&*&%\"xGF'-%%sqrtG6#F0F'F'F'!\"\"/F0;F&
\"\"%F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "`` = 2*Pi*Int(``(x^(3/2)-1),x = 2 .. 4);" "6#/%!G*(\"\"#\"\"\"%#
PiGF'-%$IntG6$-F$6#,&)%\"xG*&\"\"$F'F&!\"\"F'F'F3/F0;F&\"\"%F'" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` =
 2*Pi*(2*x^(5/2)/5-x);" "6#/%!G*(\"\"#\"\"\"%#PiGF',&*(F&F')%\"xG*&\"
\"&F'F&!\"\"F'F.F/F'F,F/F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``;" "6#%!
G" }{XPPEDIT 18 0 "PIECEWISE([4, ``],[``, ``],[2, ``]);" "6#-%*PIECEWI
SEG6%7$\"\"%%!G7$F(F(7$\"\"#F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+
2*Pi*(``(64/5-4)-(8*sqrt(2)/5-2));" "6#/%!G*(\"\"#\"\"\"%#PiGF',&-F$6#
,&*&\"#kF'\"\"&!\"\"F'\"\"%F0F',&*(\"\")F'-%%sqrtG6#F&F'F/F0F'F&F0F0F'
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*(54/5-8*sqrt(2)/5);" "6#/%!G*
(\"\"#\"\"\"%#PiGF',&*&\"#aF'\"\"&!\"\"F'*(\"\")F'-%%sqrtG6#F&F'F,F-F-
F'" }{XPPEDIT 18 0 "`` = ``(27-4*sqrt(2));" "6#/%!G-F$6#,&\"#F\"\"\"*&
\"\"%F)-%%sqrtG6#\"\"#F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4*Pi/5;
" "6#*(\"\"%\"\"\"%#PiGF%\"\"&!\"\"" }{TEXT -1 1 " " }{TEXT 338 1 "~" 
}{TEXT -1 14 " 53.64117592. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "f := x -> sqrt(x):\n'f(x)'=
f(x);\ng := x -> 1/x:\n'g(x)'=g(x);\n2*Pi*Int(x*('f(x)-g(x)'),x=2..4);
\n``=2*Pi*Int(x*(f(x)-g(x)),x=2..4);\n``=value(rhs(%));\nevalf(evalf[1
4](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$F'#\"\"\"\"
\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&\"\"\"F)F'!\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&%
\"xGF&,&-%\"fG6#F,F&-%\"gGF0!\"\"F&/F,;F%\"\"%F&F&" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\"\"%#PiGF(-%$IntG6$*&%\"xGF(,&*$F.#F(
F'F(*&F(F(F.!\"\"F3F(/F.;F'\"\"%F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/%!G,$*(\"\"#\"\"\"%#PiGF(,&#\"#a\"\"&F(*(\"\")F(F-!\"\"F'#F(F'F0F(F
(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+#f<TO&!\")" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" 
{TEXT 287 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 89 
"Find the volume of the solid of revolution formed when the region bou
nded by the curves  " }{XPPEDIT 18 0 "y = x^2+1;" "6#/%\"yG,&*$%\"xG\"
\"#\"\"\"F)F)" }{TEXT -1 6 "  and " }{XPPEDIT 18 0 "y = 3-x^2;" "6#/%
\"yG,&\"\"$\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 24 "  is rotated around \+
the " }{TEXT 286 1 "x" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT 
288 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 826 "f := x -> 3-x^2:\ng := x ->
 x^2+1:\np1 := plot([f(x),-f(x),g(x),-g(x)],x=-1.6..1.6,y=-3.3..3.3,\n
   color=[red$2,blue$2],thickness=[2,1,2,1],linestyle=[1,2,1,2]):\na :
= -1: b := 1:\npp := plot(f(x),x=a..b,adaptive=false,numpoints=25):\nf
pts := op(1,op(1,pp)):\npp := plot(g(x),x=a..b,adaptive=false,numpoint
s=25):\ngpts := op(1,op(1,pp)):\np2 := plots[polygonplot]([seq([fpts[i
-1],fpts[i],gpts[i],gpts[i-1]],i=2..25)],\n           style=patchnogri
d,color=COLOR(RGB,.95,.95,.95)):\np3 := plot([[[a,g(a)],[a,f(a)]],[[b,
g(b)],[b,f(b)]],\n      [[a,0],[a,g(a)]],[[b,0],[b,g(b)]]],linestyle=[
1$2,2$2],color=black):\nh := evalf(Pi/20): r := .07:\np4 := plot([[seq
([a+r*f(a)*cos(i*h),f(a)*sin(i*h)],i=0..40)],\n       [seq([b+r*f(b)*c
os(i*h),f(b)*sin(i*h)],i=0..40)]],\n       linestyle=2,color=black):\n
plots[display]([p1,p2,p3,p4],tickmarks=[3,7]);" }}{PARA 13 "" 1 "" 
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TG-%%VIEWG6$;$!#;F_hm$\"#;F_hm;$!#LF_hm$\"#LF_hm" 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" "Cu
rve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "C
urve 11" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 
0 "f(x) = 3-x^2;" "6#/-%\"fG6#%\"xG,&\"\"$\"\"\"*$F'\"\"#!\"\"" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x) = x^2+1;" "6#/-%\"gG6#%\"xG,&*
$F'\"\"#\"\"\"F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "The
n " }{XPPEDIT 18 0 "f(x)^2-g(x)^2=(3-x^2)^2-(x^2+1)^2" "6#/,&*$-%\"fG6
#%\"xG\"\"#\"\"\"*$-%\"gG6#F)F*!\"\",&*$,&\"\"$F+*$F)F*F0F*F+*$,&*$F)F
*F+F+F+F*F0" }{XPPEDIT 18 0 "``=9-6*x^2+x^4-(x^4+2*x^2+1)" "6#/%!G,*\"
\"*\"\"\"*&\"\"'F'*$%\"xG\"\"#F'!\"\"*$F+\"\"%F',(*$F+F/F'*&F,F'*$F+F,
F'F'F'F'F-" }{XPPEDIT 18 0 "``=8-8*x^2" "6#/%!G,&\"\")\"\"\"*&F&F'*$%
\"xG\"\"#F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "The \+
graphs of  " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT 
-1 17 " intersect where " }{XPPEDIT 18 0 "3-x^2=x^2+1" "6#/,&\"\"$\"\"
\"*$%\"xG\"\"#!\"\",&*$F(F)F&F&F&" }{TEXT -1 16 ", that is where " }
{XPPEDIT 18 0 "2*x^2=2" "6#/*&\"\"#\"\"\"*$%\"xGF%F&F%" }{TEXT -1 8 " \+
giving " }{XPPEDIT 18 0 "x =`` " "6#/%\"xG%!G" }{TEXT 290 1 "+" }
{TEXT -1 30 " 1.  \nThe required volume is: " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "V[x] = Pi*Int([f(x)^2-g(x)^2],x = -1 ..
 1);" "6#/&%\"VG6#%\"xG*&%#PiG\"\"\"-%$IntG6$7#,&*$-%\"fG6#F'\"\"#F**$
-%\"gG6#F'F4!\"\"/F';,$F*F9F*F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*Int(``(8-8*x^2),x = -1 .. 1);" 
"6#/%!G*&%#PiG\"\"\"-%$IntG6$-F$6#,&\"\")F'*&F.F'*$%\"xG\"\"#F'!\"\"/F
1;,$F'F3F'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 8*Pi*Int(``(1-x^2),x = -1 .. 1);" "6#/%!G*(\"\")\"
\"\"%#PiGF'-%$IntG6$-F$6#,&F'F'*$%\"xG\"\"#!\"\"/F0;,$F'F2F'F'" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` =
 8*Pi*(x-x^3/3);" "6#/%!G*(\"\")\"\"\"%#PiGF',&%\"xGF'*&F*\"\"$F,!\"\"
F-F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``;" "6#%!G" }{XPPEDIT 18 0 "PIE
CEWISE([1, ``],[-1, ``]);" "6#-%*PIECEWISEG6$7$\"\"\"%!G7$,$F'!\"\"F(
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 8*Pi*(``(1-1/3)-(-1+1/3));" "6#/%!
G*(\"\")\"\"\"%#PiGF',&-F$6#,&F'F'*&F'F'\"\"$!\"\"F/F',&F'F/*&F'F'F.F/
F'F/F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 8*Pi*(2-2/3);" "6#/%!G*(\"
\")\"\"\"%#PiGF',&\"\"#F'*&F*F'\"\"$!\"\"F-F'" }{XPPEDIT 18 0 "`` = 32
*Pi/3;" "6#/%!G*(\"#K\"\"\"%#PiGF'\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 
289 1 "~" }{TEXT -1 14 " 33.51032164. " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "f := x -> 3-x^2;\ng := \+
x -> x^2+1;\nPi*Int(f(x)^2-g(x)^2,x=-1..1);\nsimplify(%);\nvalue(%);\n
evalf(evalf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG
6\"6$%)operatorG%&arrowGF(,&\"\"$\"\"\"*$)9$\"\"#F.!\"\"F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow
GF(,&*$)9$\"\"#\"\"\"F1F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*
&%#PiG\"\"\"-%$IntG6$,&*$),&\"\"$F%*$)%\"xG\"\"#F%!\"\"F1F%F%*$),&F.F%
F%F%F1F%F2/F0;F2F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\")\"\"
\"%#PiGF&-%$IntG6$,&F&!\"\"*$)%\"xG\"\"#F&F&/F/;F,F&F&F," }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,$*(\"#K\"\"\"\"\"$!\"\"%#PiGF&F&" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"+k@.^L!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 
0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT 292 8 "Question
" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 89 "Find the volume of t
he solid of revolution formed when the region bounded by the curves  \+
" }{XPPEDIT 18 0 "y = sqrt(4-x^2);" "6#/%\"yG-%%sqrtG6#,&\"\"%\"\"\"*$
%\"xG\"\"#!\"\"" }{TEXT -1 6 "  and " }{XPPEDIT 18 0 "y = 1;" "6#/%\"y
G\"\"\"" }{TEXT -1 24 "  is rotated around the " }{TEXT 291 1 "x" }
{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 
112 ": The resulting solid is a sphere of radius 2 which has a circula
r hole of radius 1 drilled through its centre. " }}{PARA 0 "" 0 "" 
{TEXT 293 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 869 "f := x -> sqrt(4-x^2):
\ng := x -> 1:\np1 := plot([f(x),-f(x),g(x),-g(x)],x=-1.85..1.85,y=-2.
1..2.1,\n   color=[red$2,blue$2],thickness=[2,1,2,1],linestyle=[1,2,1,
2]):\na := -sqrt(3): b := sqrt(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)):\np2 := plots[poly
gonplot]([seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2..25)],\n      \+
     style=patchnogrid,color=COLOR(RGB,.95,.95,.95)):\np3 := plot([[[a
,g(a)],[a,f(a)]],[[b,g(b)],[b,f(b)]],\n      [[a,0],[a,g(a)]],[[b,0],[
b,g(b)]]],linestyle=[1$2,2$2],color=black):\nh := evalf(Pi/20): r := .
15:\np4 := plot([[seq([a+r*f(a)*cos(i*h),f(a)*sin(i*h)],i=0..40)],\n  \+
     [seq([b+r*f(b)*cos(i*h),f(b)*sin(i*h)],i=0..40)]],\n       linest
yle=2,color=black):\nplots[display]([p1,p2,p3,p4],tickmarks=[3,5],view
=[-2.1..2.1,-2.1..2.1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 391 445 445 
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*\\YMk:F-7$$!35x)ov!30#)=F*$!3!)*****\\hn?5%!#F7$F]bn$!31+++eYMk:F-7$$
!37x)o\\G4Z(=F*$!3A+++W*p,4$F-7$$!3;x)ohy,d'=F*$!3s*****4+0*RXF-7$$!3C
x)omN.M&=F*$!3a+++HD&y(eF-7$$!3Fx)oY#o6Q=F*$!3#)******=y1rqF-7$$!3?x)o
`f=-#=F*$!3%)*****f%*p,4)F-7$$!3Bx)[CQ\\,!=F*$!3_*****\\Cl+\"*)F-7$$!3
Gx)ylN.%y<F*$!38+++m^c5&*F-7$$!35x)os(f^b<F*$!3#******fS$)o()*F-7$$!3e
!ewu!30K<F*Fj[m7$$!3Gx)owj&e3<F*$!3w*****RS$)o()*F-7$$!35x)y$e#)p&o\"F
*$!3z*****>;l0^*F-7$$!39x)GDB_Rm\"F*$!39+++R_15*)F-7$$!3<x)='>I)Qk\"F*
$!3G+++Q*p,4)F-7$F[^n$!3=+++5y1rqF-7$$!39x)o$e#)p5;F*$!3s******=D&y(eF
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*H$*p,4$F-7$Fg\\n$!35+++XYMk:F-7$Fd\\n$\"3g******H_8/#)FfbnF`[nF^hl-F$
6%7K7$$\"35x)ov!30#)=F*F]]l7$$\"3Ix)o'eS?!)=F*Fi\\n7$$\"37x)o\\G4Z(=F*
F^]n7$$\"3;x)ohy,d'=F*Fc]n7$$\"3Cx)omN.M&=F*Fh]n7$$\"3Fx)oZ#o6Q=F*F]^n
7$$\"3?x))R&f=-#=F*Fb^n7$$\"3Cx))[#Q\\,!=F*Fg^n7$$\"3Gx)QmN.%y<F*F\\_n
7$$\"35x)[t(f^b<F*Fa_n7$$\"3e6\"Qv!30K<F*Fchl7$$\"3Gx)Gxj&e3<F*Fi_n7$$
\"35x)Q%e#)p&o\"F*F^`n7$$\"39x))eKA&Rm\"F*Fc`n7$$\"3=x)o'>I)Qk\"F*Fh`n
7$$\"36x)o.z%)fi\"F*F]^n7$$\"39x)o$e#)p5;F*F`an7$$\"3?x)o)G)*R)f\"F*Fe
an7$$\"3Ex)o+L#R*e\"F*Fjan7$$\"33x)okb(*Qe\"F*F_bn7$$\"3Gx)ov!30#e\"F*
Fdbn7$F^\\oFhbn7$$\"3Ex)o,L#R*e\"F*F]cn7$$\"3Ax)o*G)*R)f\"F*Fbcn7$$\"3
:x)o%e#)p5;F*Fgcn7$$\"37x)o/z%)fi\"F*F\\dn7$$\"3>x)o(>I)Qk\"F*Fadn7$$
\"39x))oKA&Rm\"F*Ffdn7$$\"35x)e&e#)p&o\"F*F[en7$$\"3Hx)oyj&e3<F*F`en7$
$\"3\"Q<hw!30K<F*Fj[m7$$\"37x)ou(f^b<F*Fhen7$$\"3Gx)enN.%y<F*F]fn7$$\"
3Dx)3EQ\\,!=F*Fbfn7$$\"3@x)=bf=-#=F*Fgfn7$FdinFjfn7$$\"3Cx)onN.M&=F*F_
gn7$$\"3;x)oiy,d'=F*Fdgn7$$\"37x)o]G4Z(=F*Fign7$FhhnF\\hn7$FehnF_hnF`[
nF^hl-%*AXESTICKSG6$\"\"$\"\"&-%+AXESLABELSG6%Q\"x6\"Q\"yF``o-%%FONTG6
#%(DEFAULTG-%%VIEWG6$;$!#@F[\\m$\"#@F[\\mFi`o" 1 2 0 1 10 0 2 9 1 4 1 
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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(
x) = sqrt(4-x^2);" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&\"\"%\"\"\"*$F'\"\"#!
\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x) = 1;" "6#/-%\"gG6#%\"xG
\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }
{XPPEDIT 18 0 "f(x)^2-g(x)^2 = ``(sqrt(4-x^2))^2-1^2;" "6#/,&*$-%\"fG6
#%\"xG\"\"#\"\"\"*$-%\"gG6#F)F*!\"\",&*$-%!G6#-%%sqrtG6#,&\"\"%F+*$F)F
*F0F*F+*$F+F*F0" }{XPPEDIT 18 0 "`` = 4-x^2-1;" "6#/%!G,(\"\"%\"\"\"*$
%\"xG\"\"#!\"\"F'F+" }{XPPEDIT 18 0 "`` = 3-x^2;" "6#/%!G,&\"\"$\"\"\"
*$%\"xG\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "The
 graphs of  " }{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 17 " intersect where " }{XPPEDIT 18 0 "sqrt(4-x^2) = 1;" "6#/-%%sqr
tG6#,&\"\"%\"\"\"*$%\"xG\"\"#!\"\"F)" }{TEXT -1 16 ", that is where " 
}{XPPEDIT 18 0 "4-x^2 = 1;" "6#/,&\"\"%\"\"\"*$%\"xG\"\"#!\"\"F&" }
{TEXT -1 4 " or " }{XPPEDIT 18 0 "x^2=3" "6#/*$%\"xG\"\"#\"\"$" }
{TEXT -1 9 ", giving " }{XPPEDIT 18 0 "x =`` " "6#/%\"xG%!G" }{TEXT 
294 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(3)" "6#-%%sqrtG6#\"\"$
" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "The required volume \+
is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V[x] = Pi*Int
([f(x)^2-g(x)^2],x = -sqrt(3) .. sqrt(3));" "6#/&%\"VG6#%\"xG*&%#PiG\"
\"\"-%$IntG6$7#,&*$-%\"fG6#F'\"\"#F**$-%\"gG6#F'F4!\"\"/F';,$-%%sqrtG6
#\"\"$F9-F>6#F@F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*Int(``(3-x^2)
,x = -sqrt(3) .. sqrt(3));" "6#/%!G*&%#PiG\"\"\"-%$IntG6$-F$6#,&\"\"$F
'*$%\"xG\"\"#!\"\"/F0;,$-%%sqrtG6#F.F2-F76#F.F'" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = Pi*(3*x-x^3/3);" "6#/%!G*&%#PiG\"\"\",&*&\"\"$F'%
\"xGF'F'*&F+F*F*!\"\"F-F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``;" "6#%!G
" }{XPPEDIT 18 0 "PIECEWISE([sqrt(3), ``],[``, ``],[-sqrt(3), ``]);" "
6#-%*PIECEWISEG6%7$-%%sqrtG6#\"\"$%!G7$F+F+7$,$-F(6#F*!\"\"F+" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "`` = Pi*(``(3*sqrt(3)-sqrt(3))-(-3*sqrt(3)+sqrt(
3)));" "6#/%!G*&%#PiG\"\"\",&-F$6#,&*&\"\"$F'-%%sqrtG6#F-F'F'-F/6#F-!
\"\"F',&*&F-F'-F/6#F-F'F3-F/6#F-F'F3F'" }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
``=4*sqrt(3)*Pi" "6#/%!G*(\"\"%\"\"\"-%%sqrtG6#\"\"$F'%#PiGF'" }{TEXT 
-1 1 " " }{TEXT 295 1 "~" }{TEXT -1 14 " 21.76559237. " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "f := x
 -> sqrt(4-x^2);\ng := x -> 1;\nPi*Int(f(x)^2-g(x)^2,x=-sqrt(3)..sqrt(
3));\nsimplify(%);\nvalue(%);\nevalf(evalf(%,15));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&
\"\"%\"\"\"*$)9$\"\"#F1!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"gG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#PiG\"\"\"-%$IntG6$
,&\"\"$F%*$)%\"xG\"\"#F%!\"\"/F-;,$*$F*#F%F.F/F3F%" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$*&%#PiG\"\"\"-%$IntG6$,&\"\"$!\"\"*$)%\"xG\"\"#F&F&
/F/;,$*$F+#F&F0F,F4F&F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"%\"
\"\"%#PiGF&\"\"$#F&\"\"#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+P#fl
<#!\")" }}}{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 347 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 
"" 0 "" {TEXT -1 89 "Find the volume of the solid of revolution formed
 when the region bounded by the curves  " }{XPPEDIT 18 0 "y = (12-x)/3
;" "6#/%\"yG*&,&\"#7\"\"\"%\"xG!\"\"F(\"\"$F*" }{TEXT -1 6 "  and " }
{XPPEDIT 18 0 "y = x^2/4;" "6#/%\"yG*&%\"xG\"\"#\"\"%!\"\"" }{TEXT -1 
16 ", and the lines " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 23 " is
 rotated around the " }{TEXT 349 1 "y" }{TEXT -1 8 " axis.  " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 348 8 "Solution" }
{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1573 "f := x -> (12-x)/3:\ng := x -> x^2/4:\np1 := p
lot([f(x),g(x)],x=0..4,y=-.3..4.5,\n   color=[red,blue],thickness=2,li
nestyle=1):\np2 := plot([f(-x),g(-x)],x=-4..0,y=-.3..4.5,\n   color=[r
ed,blue],thickness=1,linestyle=2):\na := 1: b := 3: c := 1.7:\npp := p
lot(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,o
p(1,pp)):\np3 := plots[polygonplot]([seq([fpts[i-1],fpts[i],gpts[i],gp
ts[i-1]],i=2..25)],\n           style=patchnogrid,color=COLOR(RGB,.9,.
9,.9)):\npp := plot(f(-x),x=-b..-a,adaptive=false,numpoints=25):\nfpts
 := op(1,op(1,pp)):\npp := plot(g(-x),x=-b..-a,adaptive=false,numpoint
s=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=patchnogri
d,color=COLOR(RGB,.92,.92,.92)):\np5 := plot([[[a,g(a)],[a,f(a)]],[[b,
g(b)],[b,f(b)]],\n      [[a,0],[a,g(a)]],[[b,0],[b,g(b)]],\n    [[-a,0
],[-a,f(a)]],[[-b,0],[-b,f(b)]]],linestyle=[1$2,2$4],color=black):\nh \+
:= evalf(Pi/20): r := .12:\np6 := plot([[seq([a*cos(i*h),f(a)+r*a*sin(
i*h)],i=0..40)],\n       [seq([a*cos(i*h),g(a)+r*a*sin(i*h)],i=0..40)]
,\n       [seq([b*cos(i*h),f(b)+r*b*sin(i*h)],i=0..40)],\n       [seq(
[b*cos(i*h),g(b)+r*b*sin(i*h)],i=0..40)]],\n       linestyle=2,color=b
lack):\np7 := plot([[seq([c*cos(i*h),f(c)+r*c*sin(i*h)],i=0..40)],\n  \+
         [seq([c*cos(i*h),g(c)+r*c*sin(i*h)],i=0..40)],\n           [[
c,g(c)],[c,f(c)]],[[-c,g(c)],[-c,f(c)]]],\n       linestyle=1,color=CO
LOR(RGB,.7,.7,.7),thickness=2): \nplots[display]([p1,p2,p3,p4,p5,p6,p7
],tickmarks=[5,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 651 308 308 
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\\r$\"3[+++-++DsF6FierF^[lFb[l-F$6&7$F`frFa\\rFierF^[lFb[l-F$6&7$7$Fda
rFafrFcarFierF^[lFb[l-%*AXESTICKSG6$\"\"&F][o-%+AXESLABELSG6%Q\"x6\"Q
\"yFd^s-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F_elF*;$F`\\oFf[o$\"#XFf[o" 1 2 
0 1 10 0 2 9 1 4 2 1.000000 46.000000 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" "C
urve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" }}}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = (12-x)/3;" "6
#/-%\"fG6#%\"xG*&,&\"#7\"\"\"F'!\"\"F+\"\"$F," }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "g(x) = x^2/4;" "6#/-%\"gG6#%\"xG*&F'\"\"#\"\"%!\"\"" }
{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 49 "Using cylindrical shel
ls the required volume is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "V[y] = 2*Pi*Int(x*(f(x)-g(x)),x = 1 .. 3);" "6#/&%\"VG6
#%\"yG*(\"\"#\"\"\"%#PiGF*-%$IntG6$*&%\"xGF*,&-%\"fG6#F0F*-%\"gG6#F0!
\"\"F*/F0;F*\"\"$F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "`` = 2*Pi*Int(x*((12-x)/3-x^2/4),x = 1 .. 3);" "6#/%
!G*(\"\"#\"\"\"%#PiGF'-%$IntG6$*&%\"xGF',&*&,&\"#7F'F-!\"\"F'\"\"$F2F'
*&F-F&\"\"%F2F2F'/F-;F'F3F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*Int(x*(4-x/3-x^2/4),x = 1 .. \+
3);" "6#/%!G*(\"\"#\"\"\"%#PiGF'-%$IntG6$*&%\"xGF',(\"\"%F'*&F-F'\"\"$
!\"\"F2*&F-F&F/F2F2F'/F-;F'F1F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*Int(``(4*x-x^2/3-x^3/4),x = 1
 .. 3);" "6#/%!G*(\"\"#\"\"\"%#PiGF'-%$IntG6$-F$6#,(*&\"\"%F'%\"xGF'F'
*&F1F&\"\"$!\"\"F4*&F1F3F0F4F4/F1;F'F3F'" }{TEXT -1 1 " " }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*(2*x^2-x^3/9-x^4/16);
" "6#/%!G*(\"\"#\"\"\"%#PiGF',(*&F&F'*$%\"xGF&F'F'*&F,\"\"$\"\"*!\"\"F
0*&F,\"\"%\"#;F0F0F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``;" "6#%!G" }
{XPPEDIT 18 0 "PIECEWISE([3, ``],[``, ``],[1, ``]);" "6#-%*PIECEWISEG6
%7$\"\"$%!G7$F(F(7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*(
``(18-3-81/16)-(2-1/9-1/16));" "6#/%!G*(\"\"#\"\"\"%#PiGF',&-F$6#,(\"#
=F'\"\"$!\"\"*&\"#\")F'\"#;F/F/F',(F&F'*&F'F'\"\"*F/F/*&F'F'F2F/F/F/F'
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*(``(10-1/16)-``(17/9-1/16));
" "6#/%!G*(\"\"#\"\"\"%#PiGF',&-F$6#,&\"#5F'*&F'F'\"#;!\"\"F0F'-F$6#,&
*&\"#<F'\"\"*F0F'*&F'F'F/F0F0F0F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+
2*Pi*(10-17/9);" "6#/%!G*(\"\"#\"\"\"%#PiGF',&\"#5F'*&\"#<F'\"\"*!\"\"
F.F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*73/9;" "6#/%!G**\"\"#\"
\"\"%#PiGF'\"#tF'\"\"*!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=146*Pi/9
" "6#/%!G*(\"$Y\"\"\"\"%#PiGF'\"\"*!\"\"" }{TEXT -1 1 " " }{TEXT 350 
1 "~" }{TEXT -1 14 " 50.96361416. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "f := x -> (12-x)/3:\n'f(x)'
=f(x);\ng := x -> x^2/4:\n'g(x)'=g(x);\n2*Pi*Int(x*('f(x)-g(x)'),x=1..
3);\n``=2*Pi*Int(x*(f(x)-g(x)),x=1..3);\n``=value(rhs(%));\nevalf(eval
f[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&\"\"%\"
\"\"*&\"\"$!\"\"F'F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"
xG,$*&\"\"%!\"\"F'\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(
\"\"#\"\"\"%#PiGF&-%$IntG6$*&%\"xGF&,&-%\"fG6#F,F&-%\"gGF0!\"\"F&/F,;F
&\"\"$F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\"\"%#PiG
F(-%$IntG6$*&%\"xGF(,(\"\"%F(*&#F(\"\"$F(F.F(!\"\"*&#F(F0F(*$)F.F'F(F(
F4F(/F.;F(F3F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"$Y\"\"\"
\"\"\"*!\"\"%#PiGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+;9O'4
&!\")" }}}{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 351 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 
"" 0 "" {TEXT -1 89 "Find the volume of the solid of revolution formed
 when 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 9 " between " }
{XPPEDIT 18 0 "x=Pi/4" "6#/%\"xG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 5 "
 and " }{XPPEDIT 18 0 "x=5*Pi/4" "6#/%\"xG*(\"\"&\"\"\"%#PiGF'\"\"%!\"
\"" }{TEXT -1 23 " is rotated around the " }{TEXT 353 1 "y" }{TEXT -1 
7 " axis. " }}{PARA 0 "" 0 "" {TEXT 352 8 "Solution" }{TEXT -1 2 ": " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 1765 "f := x -> sin(x):\ng := x -> cos(x):\np1 := plot([f(x),g(x)],x
=0..5,y=-1.2..1.2,\n   color=[red,blue],thickness=2,linestyle=1):\np2 \+
:= plot([f(-x),g(-x)],x=-5..0,y=-1.2..1.2,\n   color=[red,blue],thickn
ess=1,linestyle=2):\na := evalf(Pi/4): b := evalf(5*Pi/4): c := 2.5:\n
pp := 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)):\np3 := plots[polygonplot]([seq([fpts[i-1],fpts[i],gpt
s[i],gpts[i-1]],i=2..25)],\n           style=patchnogrid,color=COLOR(R
GB,.9,.9,.9)):\npp := plot(f(-x),x=-b..-a,adaptive=false,numpoints=25)
:\nfpts := op(1,op(1,pp)):\npp := plot(g(-x),x=-b..-a,adaptive=false,n
umpoints=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=pat
chnogrid,color=COLOR(RGB,.92,.92,.92)):\np5 := plot([[[a,g(a)],[a,f(a)
]],[[b,g(b)],[b,f(b)]],\n      [[a,0],[a,g(a)]],[[b,0],[b,g(b)]],\n   \+
 [[-a,0],[-a,f(a)]],[[-b,0],[-b,f(b)]]],linestyle=[1$2,2$4],color=blac
k):\nh := evalf(Pi/20): r := .12:\np6 := plot([[seq([a*cos(i*h),f(a)+r
*a*sin(i*h)],i=0..40)],\n       [seq([a*cos(i*h),g(a)+r*a*sin(i*h)],i=
0..40)],\n       [seq([b*cos(i*h),f(b)+r*b*sin(i*h)],i=0..40)],\n     \+
  [seq([b*cos(i*h),g(b)+r*b*sin(i*h)],i=0..40)]],\n       linestyle=2,
color=black):\np7 := plot([[seq([c*cos(i*h),f(c)+r*c*sin(i*h)],i=0..40
)],\n           [seq([c*cos(i*h),g(c)+r*c*sin(i*h)],i=0..40)],\n      \+
     [[c,g(c)],[c,f(c)]],[[-c,g(c)],[-c,f(c)]]],\n       linestyle=1,c
olor=COLOR(RGB,.7,.7,.7),thickness=2): \nplots[display]([p1,p2,p3,p4,p
5,p6,p7],ytickmarks=3,labels=[``,``],\n  xtickmarks=[evalf(-3*Pi/2)=`-
3p/2`,evalf(-Pi)=`-p`,evalf(-Pi/2)=`-p/2`,\n    evalf(Pi/2)=`p/2`,eval
f(Pi)=`p`,evalf(3*Pi/2)=`3p/2`],\n      font=[SYMBOL,10]);" }}{PARA 
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Fjjr$!3-+++=,X(4\"Fen7$F_[s$!35+++;O9,6Fen7$Fd[sFfcs7$Fi[sFccs7$F^\\sF
`cs7$Fc\\sF]cs7$F\\drFjbs7$F[]s$!3M+++6>zu(*F-7$F`]s$!3!)*****>l2MP*F-
7$Fe]s$!3Z*****\\8([Q*)F-7$Fhbr$!3I+++[&R2[)F-7$Fcbr$!3,+++`hV6!)F-F_^
sF^[lFb[l-F$6&7$Ff^sFbbrF_^sF^[lFb[l-F$6&7$7$F_hrFg^s7$F_hrFebrF_^sF^[
lFb[l-%+AXESLABELSG6%%!GFjes-%%FONTG6#%(DEFAULTG-F\\fs6$%'SYMBOLG\"#5-
%*AXESTICKSG6$7(/$!+\")*)Q7ZFj^n%&-3p/2G/$!+aEfTJFj^n%#-pG/$!+Fjzq:Fj^
n%%-p/2G/$\"+Fjzq:Fj^n%$p/2G/$\"+aEfTJFj^n%\"pG/$\"+\")*)Q7ZFj^n%%3p/2
G\"\"$-%%VIEWG6$;FaelFcz;$FdanF\\\\o$\"#7F\\\\o" 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" "Curv
e 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Cur
ve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 1
7" "Curve 18" "Curve 19" "Curve 20" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = sin*x;" "6#/-%\"fG6#%\"xG*&%$
sinG\"\"\"F'F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x) = cos*x;" "6#
/-%\"gG6#%\"xG*&%$cosG\"\"\"F'F*" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" 
{TEXT -1 49 "Using cylindrical shells the required volume is: " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V[y] = 2*Pi*Int(x*(f(
x)-g(x)),x = Pi/4 .. 5*Pi/4);" "6#/&%\"VG6#%\"yG*(\"\"#\"\"\"%#PiGF*-%
$IntG6$*&%\"xGF*,&-%\"fG6#F0F*-%\"gG6#F0!\"\"F*/F0;*&F+F*\"\"%F8*(\"\"
&F*F+F*F<F8F*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 2*Pi*Int(x*(sin*x-cos*x),x = Pi/4 .. 5*Pi/4);" "6#
/%!G*(\"\"#\"\"\"%#PiGF'-%$IntG6$*&%\"xGF',&*&%$sinGF'F-F'F'*&%$cosGF'
F-F'!\"\"F'/F-;*&F(F'\"\"%F3*(\"\"&F'F(F'F7F3F'" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 69 "This integral can be found by using the i
ntegration by parts formula " }{XPPEDIT 18 0 "Int(u*``(dv/dx),x)=u*v-I
nt(v*``(du/dx),x)" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"
F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT 
-1 8 "  with: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "PI
ECEWISE([u=x,v=-cos*x-sin*x],[du/dx=1,dv/dx=sin*x-cos*x])" "6#-%*PIECE
WISEG6$7$/%\"uG%\"xG/%\"vG,&*&%$cosG\"\"\"F)F/!\"\"*&%$sinGF/F)F/F07$/
*&%#duGF/%#dxGF0F//*&%#dvGF/F7F0,&*&F2F/F)F/F/*&F.F/F)F/F0" }{TEXT -1 
1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "V[y] = 2*Pi*(-cos*x-sin*x)*x;" "6#/&%\"VG6#%\"yG**
\"\"#\"\"\"%#PiGF*,&*&%$cosGF*%\"xGF*!\"\"*&%$sinGF*F/F*F0F*F/F*" }
{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 " " }{XPPEDIT 18 0 "``-2*Pi*Int(``(-cos*x-sin*x),x = Pi/4 .
. 5*Pi/4);" "6#,&%!G\"\"\"*(\"\"#F%%#PiGF%-%$IntG6$-F$6#,&*&%$cosGF%%
\"xGF%!\"\"*&%$sinGF%F1F%F2/F1;*&F(F%\"\"%F2*(\"\"&F%F(F%F8F2F%F2" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` =
 -2*Pi*x*(cos*x+sin*x);" "6#/%!G,$**\"\"#\"\"\"%#PiGF(%\"xGF(,&*&%$cos
GF(F*F(F(*&%$sinGF(F*F(F(F(!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIEC
EWISE([5*Pi/4,``],[Pi/4,``])" "6#-%*PIECEWISEG6$7$*(\"\"&\"\"\"%#PiGF)
\"\"%!\"\"%!G7$*&F*F)F+F,F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``+2*Pi*I
nt(``(cos*x+sin*x),x = Pi/4 .. 5*Pi/4);" "6#,&%!G\"\"\"*(\"\"#F%%#PiGF
%-%$IntG6$-F$6#,&*&%$cosGF%%\"xGF%F%*&%$sinGF%F1F%F%/F1;*&F(F%\"\"%!\"
\"*(\"\"&F%F(F%F7F8F%F%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "`` = -2*Pi*x*(cos*x+sin*x);" "6#/%!G,$**\"\"#\"
\"\"%#PiGF(%\"xGF(,&*&%$cosGF(F*F(F(*&%$sinGF(F*F(F(F(!\"\"" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "PIECEWISE([5*Pi/4,``],[Pi/4,``])" "6#-%*PIECE
WISEG6$7$*(\"\"&\"\"\"%#PiGF)\"\"%!\"\"%!G7$*&F*F)F+F,F-" }{TEXT -1 1 
" " }{XPPEDIT 18 0 "``+2*Pi*(sin*x-cos*x);" "6#,&%!G\"\"\"*(\"\"#F%%#P
iGF%,&*&%$sinGF%%\"xGF%F%*&%$cosGF%F,F%!\"\"F%F%" }{TEXT -1 2 "  " }
{XPPEDIT 18 0 "PIECEWISE([5*Pi/4, ``],[Pi/4, ``])" "6#-%*PIECEWISEG6$7
$*(\"\"&\"\"\"%#PiGF)\"\"%!\"\"%!G7$*&F*F)F+F,F-" }{TEXT -1 1 " " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*(-x*(cos*x+
sin*x)+sin*x-cos*x);" "6#/%!G*(\"\"#\"\"\"%#PiGF',(*&%\"xGF',&*&%$cosG
F'F+F'F'*&%$sinGF'F+F'F'F'!\"\"*&F0F'F+F'F'*&F.F'F+F'F1F'" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "PIECEWISE([5*Pi/4, ``],[Pi/4, ``])" "6#-%*PIECEW
ISEG6$7$*(\"\"&\"\"\"%#PiGF)\"\"%!\"\"%!G7$*&F*F)F+F,F-" }{TEXT -1 1 "
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " 
}{XPPEDIT 18 0 "`` = 2*Pi*(5*Pi/4*sqrt(2)-(-Pi/4*sqrt(2)));" "6#/%!G*(
\"\"#\"\"\"%#PiGF',&**\"\"&F'F(F'\"\"%!\"\"-%%sqrtG6#F&F'F',$*(F(F'F,F
--F/6#F&F'F-F-F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=3*Pi^2*sqrt(2)" "
6#/%!G*(\"\"$\"\"\"*$%#PiG\"\"#F'-%%sqrtG6#F*F'" }{TEXT -1 1 " " }
{TEXT 354 1 "~" }{TEXT -1 14 " 41.87318520. " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 185 "f := x -> sin(x):\n'f(x)'=f(x);\ng := x -> cos(x):\n
'g(x)'=g(x);\n2*Pi*Int(x*('f(x)-g(x)'),x=Pi/4..5*Pi/4);\n``=2*Pi*Int(x
*(f(x)-g(x)),x=Pi/4..5*Pi/4);\n``=value(rhs(%));\nevalf(evalf[14](%));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%$sinGF&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%$cosGF&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&%\"xGF&,&-%\"fG6#F,F&-
%\"gGF0!\"\"F&/F,;,$*&\"\"%F3F'F&F&,$*(\"\"&F&F8F3F'F&F&F&F&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\"\"%#PiGF(-%$IntG6$*&%\"xGF(
,&-%$sinG6#F.F(-%$cosGF2!\"\"F(/F.;,$*&\"\"%F5F)F(F(,$*(\"\"&F(F:F5F)F
(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"$\"\"\")%#PiG\"
\"#F(F+#F(F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+?&=t=%!\")" 
}}}{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 297 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 
0 "" {TEXT -1 111 "Find the volume of the solid of revolution formed w
hen the region in the first quadrant bounded by the curves  " }
{XPPEDIT 18 0 "y = 5-2/x;" "6#/%\"yG,&\"\"&\"\"\"*&\"\"#F'%\"xG!\"\"F+
" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "y = x^2;" "6#/%\"yG*$%\"xG\"\"
#" }{TEXT -1 23 " is rotated around the " }{TEXT 296 1 "x" }{TEXT -1 
7 " axis. " }}{PARA 0 "" 0 "" {TEXT 298 8 "Solution" }{TEXT -1 2 ": " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 851 "f := x -> 5-2/x:\ng := x -> x^2;\np1 := plot([f(x),-f(x),g(x),-
g(x)],x=0.41..2.4,y=-5..5,\n   color=[red$2,blue$2],thickness=[2,1,2,1
],linestyle=[1,2,1,2]):\na := sqrt(2)-1: b := 2:\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)):\np2
 := plots[polygonplot]([seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2.
.25)],\n           style=patchnogrid,color=COLOR(RGB,.95,.95,.95)):\np
3 := plot([[[a,g(a)],[a,f(a)]],[[b,g(b)],[b,f(b)]],\n      [[a,0],[a,g
(a)]],[[b,0],[b,g(b)]]],linestyle=[1$2,2$2],color=black):\nh := evalf(
Pi/20): r := .05:\np4 := plot([[seq([a+r*f(a)*cos(i*h),f(a)*sin(i*h)],
i=0..40)],\n       [seq([b+r*f(b)*cos(i*h),f(b)*sin(i*h)],i=0..40)]],
\n       linestyle=[1,2],color=black):\nplots[display]([p1,p2,p3,p4],v
iew=[0..2.4,-5..5],tickmarks=[3,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#\"\"\"F(F(F(" }}
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\"+1W(QZ\"Fjjm$\"+vA.VOFjjm7$Febn$\"+ldIs@FjjmF`bn7&Fdbn7$$\"+&3CW`\"F
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\"+********RFjjm7$F]gn$\"+#*******RFjjmFhfn-%&COLORG6&Fe[l$\"#&*!\"#Fg
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F*$\"3&oBM6U.K@\"F*7$$\"33Ao5E'fD>%F*$\"3#[hbPr`!)Q\"F*7$$\"3'\\439&=3
\"=%F*$\"333Py2bsG:F*7$$\"3yG_H!4X'oTF*$\"33)yZc+b<j\"F*7$$\"31HDAzbbb
TF*$\"3?,DM*G0Yp\"F*7$$\"3\"[6EUiN@9%F*Fghn7$$\"3Je]?scrGTF*$\"3E4DMz_
g%p\"F*7$$\"3QeB8_hi:TF*Fg\\o7$$\"3G%\\>gR*=.TF*$\"3sMe#>^D(G:F*7$$\"3
!*4O!Gi6<4%F*$\"3!fu(*yr`!)Q\"F*7$$\"3Wv*\\\\Xv93%F*Fh[o7$$\"3=[<!Q$Ht
sSF*$\"3?iU<n+[35F*7$$\"3qX.5\\$*plSF*$\"3/P9G![X#*y(Fejn7$$\"3#*Rr?uy
agSF*$\"3HjSU4M*=I&Fejn7$$\"3SZ_+h`SdSF*$\"3)\\<%)[4\"*Ro#Fejn7$$\"3sX
S5'=\\j0%F*$!3SK4\">SN!Qq!#G7$$\"3w/C_g`SdSF*$!3Y&**4'36*Ro#Fejn7$Fa_o
$!3ch)\\JU$*=I&Fejn7$$\"3%)fY8[$*plSF*$!3_ds+%\\X#*y(Fejn7$$\"3ojg$G%H
tsSF*$!3K!***))[+[35F*7$$\"3#[g=IXv93%F*$!3a)4#*pV.K@\"F*7$$\"3GRA(3i6
<4%F*$!3k:cvBP0)Q\"F*7$$\"3mB\")3%R*=.TF*$!333Py2bsG:F*7$$\"3/))4?ghi:
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!3T5\"QDvGdr\"F*7$$\"3ge6HzbbbTF*$!3E4DMz_g%p\"F*7$$\"3ydQO!4X'oTF*$!3
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(*yq`!)Q\"F*7$Ff[o$!3wojFNM?87F*7$$\"32oWp4$Q:@%F*$!3PhU<d+[35F*7$$\"3
t9(y)**=d=UF*$!3gpU8DbC*y(Fejn7$$\"3uw!*GuLsBUF*$!3PloFaL*=I&Fejn7$$\"
3Y7Q(z)e'oA%F*$!33EbI'3\"*Ro#Fejn7$F^jn$\"3*\\=#Qqqg29!#FF_\\lFihn-F$6
%7K7$$\"3;+++++++AFFFi[l7$$\"3#)*****zmPv>#FF$\"37+++/'ytD'F*7$$\"3))*
****HI6->#FF$\"3!******z(z1O7FF7$$\"3)******\\I,#y@FF$\"3********)*>'f
\"=FF7$$\"33+++*R.=;#FF$\"33+++559^BFF7$$\"3%******fN@99#FF$\"3:+++DrU
GGFF7$$\"3$)******\\qb<@FF$\"3!******z(z1OKFF7$$\"3?++++\")z!4#FF$\"3=
+++(4ESc$FF7$$\"3)*******)R.=1#FF$\"3-+++mgA/QFF7$$\"3()*****H*oGJ?FF$
\"3%******>O`2&RFFF^in7$$\"3#******p58(o>FFF`io7$$\"3-+++,m>Q>FF$\"3$*
*****\\1EU!QFF7$$\"3,++++>?4>FF$\"33+++'4ESc$FF7$$\"3%*******\\HW#)=FF
$\"3=+++wz1OKFF7$$\"31+++W'y&e=FFF\\ho7$$\"3$******4g'>Q=FF$\"3++++459
^BFF7$$\"3-+++&p)z@=FF$\"30+++(*>'f\"=FF7$$\"3*)*****pp)y4=FF$\"3-+++u
z1O7FF7$$\"3%******>LiC!=FF$\"3!)*******fytD'F*7$$\"3/+++++++=FF$!3#**
****f/F3k\"!#E7$Fg[p$!3A+++K'ytD'F*7$Fb[p$!3!******z(z1O7FF7$F][p$!33+
+++?'f\"=FF7$Fhjo$!3!)*****>,T6N#FF7$Fejo$!3'******z7F%GGFF7$F`jo$!3!*
*****z(z1OKFF7$F[jo$!3!)*****z4ESc$FF7$Ffio$!3-+++mgA/QFF7$Fcio$!3%***
***>O`2&RFF7$F_in$!\"%Fj[l7$F^ioFj]p7$Fiho$!3$******\\1EU!QFF7$Fdho$!3
3+++'4ESc$FF7$$\"3\"******40dv6#FF$!35+++vz1OKFF7$Fjgo$!32+++CrUGGFF7$
Fego$!3#******z+T6N#FF7$F`go$!3(******f*>'f\"=FF7$F[go$!3$******H(z1O7
FF7$Fffo$!3O+++!eytD'F*7$Fcfo$\"3%)*****>4a;G$F`\\pF`flFihn-%*AXESTICK
SG6$\"\"$\"\"&-%+AXESLABELSG6%Q\"x6\"Q\"yFf`p-%%FONTG6#%(DEFAULTG-%%VI
EWG6$;Fi[l$\"#C!\"\";$!\"&Fj[l$Fa`pFj[l" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(
x) = 5-2/x;" "6#/-%\"fG6#%\"xG,&\"\"&\"\"\"*&\"\"#F*F'!\"\"F-" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "g(x) = x^2;" "6#/-%\"gG6#%\"xG*$F'\"\"#" 
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "
f(x)^2-g(x)^2 = ``(5-2/x)^2-(x^2)^2;" "6#/,&*$-%\"fG6#%\"xG\"\"#\"\"\"
*$-%\"gG6#F)F*!\"\",&*$-%!G6#,&\"\"&F+*&F*F+F)F0F0F*F+*$*$F)F*F*F0" }
{XPPEDIT 18 0 "`` = 25-20/x+4/(x^2)-x^4;" "6#/%!G,*\"#D\"\"\"*&\"#?F'%
\"xG!\"\"F+*&\"\"%F'*$F*\"\"#F+F'*$F*F-F+" }{TEXT -1 2 ". " }}{PARA 0 
"" 0 "" {TEXT -1 15 "The graphs of  " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"y
G-%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-
%\"gG6#%\"xG" }{TEXT -1 17 " intersect where " }{XPPEDIT 18 0 "5-2/x =
 x^2;" "6#/,&\"\"&\"\"\"*&\"\"#F&%\"xG!\"\"F**$F)F(" }{TEXT -1 16 ", t
hat is where " }{XPPEDIT 18 0 "5*x-2 = x^3;" "6#/,&*&\"\"&\"\"\"%\"xGF
'F'\"\"#!\"\"*$F(\"\"$" }{TEXT -1 5 " or  " }{XPPEDIT 18 0 "x^3-5*x+2 \+
= 0;" "6#/,(*$%\"xG\"\"$\"\"\"*&\"\"&F(F&F(!\"\"\"\"#F(\"\"!" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "By inspection " }{XPPEDIT 18 
0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 41 " is a solution of this equation,
 so that " }{XPPEDIT 18 0 "x^3-5*x+2" "6#,(*$%\"xG\"\"$\"\"\"*&\"\"&F'
F%F'!\"\"\"\"#F'" }{TEXT -1 12 " factors as " }{XPPEDIT 18 0 "(x-2)*(x
^2+2*x-1)" "6#*&,&%\"xG\"\"\"\"\"#!\"\"F&,(*$F%F'F&*&F'F&F%F&F&F&F(F&
" }{TEXT -1 6 ". Now " }{XPPEDIT 18 0 "x^2+2*x-1=0" "6#/,(*$%\"xG\"\"#
\"\"\"*&F'F(F&F(F(F(!\"\"\"\"!" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "(
x-1)^2=2" "6#/*$,&%\"xG\"\"\"F'!\"\"\"\"#F)" }{TEXT -1 43 " so the zer
os of the quadradic factor are  " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"
" }{TEXT -1 1 " " }{TEXT 299 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqr
t(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 29 ". This means that the curves \+
" }{XPPEDIT 18 0 "y=5-2/x" "6#/%\"yG,&\"\"&\"\"\"*&\"\"#F'%\"xG!\"\"F+
" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"#" }
{TEXT -1 34 " meet in the first quadrant where " }{XPPEDIT 18 0 "x =sq
rt(2)-1" "6#/%\"xG,&-%%sqrtG6#\"\"#\"\"\"F*!\"\"" }{TEXT -1 5 " and " 
}{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 24 "The required volume is: " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V[x] = Pi*Int([f
(x)^2-g(x)^2],x = sqrt(2)-1 .. 2);" "6#/&%\"VG6#%\"xG*&%#PiG\"\"\"-%$I
ntG6$7#,&*$-%\"fG6#F'\"\"#F**$-%\"gG6#F'F4!\"\"/F';,&-%%sqrtG6#F4F*F*F
9F4F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*Int(``(25-20/x+4/(x^2)-x^4
),x = sqrt(2)-1 .. 2);" "6#/%!G*&%#PiG\"\"\"-%$IntG6$-F$6#,*\"#DF'*&\"
#?F'%\"xG!\"\"F2*&\"\"%F'*$F1\"\"#F2F'*$F1F4F2/F1;,&-%%sqrtG6#F6F'F'F2
F6F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*(25*x-20*ln(x)-4/x-x^5/5);
" "6#/%!G*&%#PiG\"\"\",**&\"#DF'%\"xGF'F'*&\"#?F'-%#lnG6#F+F'!\"\"*&\"
\"%F'F+F1F1*&F+\"\"&F5F1F1F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``;" "6#
%!G" }{XPPEDIT 18 0 "PIECEWISE([2, ``],[``, ``],[sqrt(2)-1, ``]);" "6#
-%*PIECEWISEG6%7$\"\"#%!G7$F(F(7$,&-%%sqrtG6#F'\"\"\"F/!\"\"F(" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*(``(50-20*ln(2)-2-32/5)-(25*(sq
rt(2)-1)-20*ln(sqrt(2)-1)-4/(sqrt(2)-1)-(sqrt(2)-1)^5/5));" "6#/%!G*&%
#PiG\"\"\",&-F$6#,*\"#]F'*&\"#?F'-%#lnG6#\"\"#F'!\"\"F2F3*&\"#KF'\"\"&
F3F3F',**&\"#DF',&-%%sqrtG6#F2F'F'F3F'F'*&F.F'-F06#,&-F<6#F2F'F'F3F'F3
*&\"\"%F',&-F<6#F2F'F'F3F3F3*&,&-F<6#F2F'F'F3F6F6F3F3F3F'" }{TEXT -1 
1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "`` = Pi*(``(208/5-20*ln(2))-(25*(sqrt(2)-1)-20*ln(s
qrt(2)-1)-4*(sqrt(2)+1)-(29*sqrt(2)-41)/5));" "6#/%!G*&%#PiG\"\"\",&-F
$6#,&*&\"$3#F'\"\"&!\"\"F'*&\"#?F'-%#lnG6#\"\"#F'F/F',**&\"#DF',&-%%sq
rtG6#F5F'F'F/F'F'*&F1F'-F36#,&-F;6#F5F'F'F/F'F/*&\"\"%F',&-F;6#F5F'F'F
'F'F/*&,&*&\"#HF'-F;6#F5F'F'\"#TF/F'F.F/F/F/F'" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = Pi*(``(208/5-20*ln(2))-((76*sqrt(2)-104)/5-20*ln(s
qrt(2)-1)));" "6#/%!G*&%#PiG\"\"\",&-F$6#,&*&\"$3#F'\"\"&!\"\"F'*&\"#?
F'-%#lnG6#\"\"#F'F/F',&*&,&*&\"#wF'-%%sqrtG6#F5F'F'\"$/\"F/F'F.F/F'*&F
1F'-F36#,&-F<6#F5F'F'F/F'F/F/F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=((
312-76*sqrt(2))/5+20*ln((sqrt(2)-1)/2))*Pi" "6#/%!G*&,&*&,&\"$7$\"\"\"
*&\"#wF*-%%sqrtG6#\"\"#F*!\"\"F*\"\"&F1F**&\"#?F*-%#lnG6#*&,&-F.6#F0F*
F*F1F*F0F1F*F*F*%#PiGF*" }{TEXT -1 1 " " }{TEXT 300 1 "~" }{TEXT -1 
14 " 29.57350340. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 129 "f := x -> 5-2/x;\ng := x -> x^2;\nPi*Int(f(
x)^2-g(x)^2,x=sqrt(2)-1..2);\ncombine(expand(rationalize(value(%))));
\nevalf(evalf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"
xG6\"6$%)operatorG%&arrowGF(,&\"\"&\"\"\"*&\"\"#F.9$!\"\"F2F(F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow
GF(*$)9$\"\"#\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#PiG\"
\"\"-%$IntG6$,&*$),&\"\"&F%*&\"\"#F%%\"xG!\"\"F1F/F%F%*$)F0\"\"%F%F1/F
0;,&*$F/#F%F/F%F%F1F/F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(**\"#w\"
\"\"\"\"&!\"\"%#PiGF&\"\"##F&F*F(*(\"$7$F&F'F(F)F&F&*(\"#?F&F)F&-%#lnG
6#,$*&F*F&,&*$F*F+F&F&F(F(F&F&F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"+S.NdH!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Examp
le 7 " }}{PARA 0 "" 0 "" {TEXT 341 8 "Question" }{TEXT -1 2 ": " }}
{PARA 0 "" 0 "" {TEXT -1 89 "Find the volume of the solid of revolutio
n formed when the region bounded by the curves  " }{XPPEDIT 18 0 "y = \+
6-x^2;" "6#/%\"yG,&\"\"'\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 12 ", the l
ine  " }{XPPEDIT 18 0 "y = x;" "6#/%\"yG%\"xG" }{TEXT -1 9 " and the \+
" }{TEXT 343 1 "y" }{TEXT -1 28 " axis is rotated around the " }{TEXT 
340 1 "x" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT 342 8 "Solutio
n" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 845 "f := x -> 6-x^2:\ng := x -> x:\np1 := plot(
[f(x),-f(x),g(x),-g(x)],x=0..2.3,y=-6.1..6.1,\n   color=[red$2,blue$2]
,thickness=[2,1,2,1],linestyle=[1,2,1,2]):\na := 0: b := 2:\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)):\np2 := plots[polygonplot]([seq([fpts[i-1],fpts[i],gpts[i],gpts
[i-1]],i=2..25)],\n           style=patchnogrid,color=COLOR(RGB,.95,.9
5,.95)):\np3 := plot([[[a,g(a)],[a,f(a)]],[[b,g(b)],[b,f(b)]],\n      \+
[[a,0],[a,g(a)]],[[b,0],[b,g(b)]]],linestyle=[1$2,2$2],color=black):\n
h := evalf(Pi/20): r := .07:\np4 := plot([[seq([a+r*f(a)*cos(i*h),f(a)
*sin(i*h)],i=0..40)],\n       [seq([b+r*f(b)*cos(i*h),f(b)*sin(i*h)],i
=0..40)]],\n       linestyle=2,color=black):\nplots[display]([p1,p2,p3
,p4],tickmarks=[3,7],view=[-.45..2.3,-6.1..6.1]);" }}{PARA 13 "" 1 "" 
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IEWG6$;$!#XFh\\n$\"#B!\"\";$!#hF`eo$\"#hF`eo" 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" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(
x) = 6-x^2;" "6#/-%\"fG6#%\"xG,&\"\"'\"\"\"*$F'\"\"#!\"\"" }{TEXT -1 
5 " and " }{XPPEDIT 18 0 "g(x) = x;" "6#/-%\"gG6#%\"xGF'" }{TEXT -1 2 
". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "f(x)^2-g(x)^
2 = (6-x^2)^2-x^2;" "6#/,&*$-%\"fG6#%\"xG\"\"#\"\"\"*$-%\"gG6#F)F*!\"
\",&*$,&\"\"'F+*$F)F*F0F*F+*$F)F*F0" }{XPPEDIT 18 0 "`` = 36-12*x^2+x^
4-x^2;" "6#/%!G,*\"#O\"\"\"*&\"#7F'*$%\"xG\"\"#F'!\"\"*$F+\"\"%F'*$F+F
,F-" }{XPPEDIT 18 0 "`` = 36-13*x^2+x^4;" "6#/%!G,(\"#O\"\"\"*&\"#8F'*
$%\"xG\"\"#F'!\"\"*$F+\"\"%F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 15 "The graphs of  " }{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 17 " intersect where " }{XPPEDIT 18 0 "6-x^2 = x;" "6#
/,&\"\"'\"\"\"*$%\"xG\"\"#!\"\"F(" }{TEXT -1 16 ", that is where " }
{XPPEDIT 18 0 "x^2+x-6 = 0;" "6#/,(*$%\"xG\"\"#\"\"\"F&F(\"\"'!\"\"\"
\"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "(x-2)*(x+3)=0" "6#/*&,&%\"xG\"
\"\"\"\"#!\"\"F',&F&F'\"\"$F'F'\"\"!" }{TEXT -1 9 ", giving " }
{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 4 " or " }{XPPEDIT 
18 0 "x=-3" "6#/%\"xG,$\"\"$!\"\"" }{TEXT -1 28 ".  \nThe required vol
ume is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V[x] = Pi
*Int([f(x)^2-g(x)^2],x = 0 .. 2);" "6#/&%\"VG6#%\"xG*&%#PiG\"\"\"-%$In
tG6$7#,&*$-%\"fG6#F'\"\"#F**$-%\"gG6#F'F4!\"\"/F';\"\"!F4F*" }{TEXT 
-1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*In
t(``(36-13*x^2+x^4),x = 0 .. 2);" "6#/%!G*&%#PiG\"\"\"-%$IntG6$-F$6#,(
\"#OF'*&\"#8F'*$%\"xG\"\"#F'!\"\"*$F2\"\"%F'/F2;\"\"!F3F'" }{TEXT -1 
1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*(36*x
-13*x^3/3+x^5/5);" "6#/%!G*&%#PiG\"\"\",(*&\"#OF'%\"xGF'F'*(\"#8F'*$F+
\"\"$F'F/!\"\"F0*&F+\"\"&F2F0F'F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``;
" "6#%!G" }{XPPEDIT 18 0 "PIECEWISE([2, ``],[``, ``],[0, ``]);" "6#-%*
PIECEWISEG6%7$\"\"#%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Pi*(72-104/3+32/5);" "6#/%!G*
&%#PiG\"\"\",(\"#sF'*&\"$/\"F'\"\"$!\"\"F-*&\"#KF'\"\"&F-F'F'" }
{XPPEDIT 18 0 "``=Pi " "6#/%!G%#PiG" }{TEXT -1 1 " " }{TEXT 345 1 "(" 
}{TEXT -1 9 " 72 - 34 " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"
\"" }{TEXT -1 5 " + 6 " }{XPPEDIT 18 0 "2/5" "6#*&\"\"#\"\"\"\"\"&!\"
\"" }{TEXT -1 1 " " }{TEXT 344 1 ")" }{TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "`` = Pi *(44-2/3+2/
5)" "6#/%!G*&%#PiG\"\"\",(\"#WF'*&\"\"#F'\"\"$!\"\"F-*&F+F'\"\"&F-F'F'
" }{XPPEDIT 18 0 "``=  Pi*(44-4/15)" "6#/%!G*&%#PiG\"\"\",&\"#WF'*&\"
\"%F'\"#:!\"\"F-F'" }{XPPEDIT 18 0 "`` = 43;" "6#/%!G\"#V" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "11/15;" "6#*&\"#6\"\"\"\"#:!\"\"" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "Pi=656*Pi/15" "6#/%#PiG*(\"$c'\"\"\"F$F'\"#:!\"\"" }
{TEXT -1 1 " " }{TEXT 346 1 "~" }{TEXT -1 14 " 137.3923187. " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "f
 := x -> 6-x^2;\ng := x -> x;\nPi*Int(f(x)^2-g(x)^2,x=0..2);\nsimplify
(%);\nvalue(%);\nevalf(evalf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"'\"\"\"*$)9$\"\"#F.!
\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)op
eratorG%&arrowGF(9$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#PiG\"
\"\"-%$IntG6$,&*$),&\"\"'F%*$)%\"xG\"\"#F%!\"\"F1F%F%F.F2/F0;\"\"!F1F%
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#PiG\"\"\"-%$IntG6$,(\"#OF%*&\"
#8F%)%\"xG\"\"#F%!\"\"*$)F.\"\"%F%F%/F.;\"\"!F/F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*(\"$c'\"\"\"\"#:!\"\"%#PiGF&F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+(=BRP\"!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 10 "Example 8 " }}{PARA 0 "" 0 "" {TEXT 355 8 "Question" }
{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 89 "Find the volume of the \+
solid of revolution formed when the region bounded by the curves  " }
{XPPEDIT 18 0 "y = arctan*x;" "6#/%\"yG*&%'arctanG\"\"\"%\"xGF'" }
{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "y = -x/(x^2+1);" "6#/%\"yG,$*&%\"
xG\"\"\",&*$F'\"\"#F(F(F(!\"\"F," }{TEXT -1 9 " between " }{XPPEDIT 
18 0 "x = 1/sqrt(3);" "6#/%\"xG*&\"\"\"F&-%%sqrtG6#\"\"$!\"\"" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "x = sqrt(3);" "6#/%\"xG-%%sqrtG6#\"\"$" }
{TEXT -1 23 " is rotated around the " }{TEXT 357 1 "y" }{TEXT -1 7 " a
xis. " }}{PARA 0 "" 0 "" {TEXT 356 8 "Solution" }{TEXT -1 2 ": " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1628 "f := x -> arctan(x):\ng := x -> -x/(x^2+1):\np1 := plot([f(x),g(
x)],x=0..2,y=-.8..1.5,\n   color=[red,blue],thickness=2,linestyle=1):
\np2 := plot([f(-x),g(-x)],x=-2..0,y=-.8..1.5,\n   color=[red,blue],th
ickness=1,linestyle=2):\na := evalf(1/sqrt(3)): b := evalf(sqrt(3)): c
 := 1.2:\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)):\np3 := plots[polygonplot]([seq([fpts[i-1],f
pts[i],gpts[i],gpts[i-1]],i=2..25)],\n           style=patchnogrid,col
or=COLOR(RGB,.9,.9,.9)):\npp := plot(f(-x),x=-b..-a,adaptive=false,num
points=25):\nfpts := op(1,op(1,pp)):\npp := plot(g(-x),x=-b..-a,adapti
ve=false,numpoints=25):\ngpts := op(1,op(1,pp)):\np4 := plots[polygonp
lot]([seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2..25)],\n          \+
 style=patchnogrid,color=COLOR(RGB,.92,.92,.92)):\np5 := plot([[[a,g(a
)],[a,f(a)]],[[b,g(b)],[b,f(b)]],\n      [[a,0],[a,g(a)]],[[b,0],[b,g(
b)]],\n    [[-a,g(a)],[-a,f(a)]],[[-b,g(b)],[-b,f(b)]]],linestyle=[1$2
,2$4],color=black):\nh := evalf(Pi/20): r := .12:\np6 := plot([[seq([a
*cos(i*h),f(a)+r*a*sin(i*h)],i=0..40)],\n       [seq([a*cos(i*h),g(a)+
r*a*sin(i*h)],i=0..40)],\n       [seq([b*cos(i*h),f(b)+r*b*sin(i*h)],i
=0..40)],\n       [seq([b*cos(i*h),g(b)+r*b*sin(i*h)],i=0..40)]],\n   \+
    linestyle=2,color=black):\np7 := plot([[seq([c*cos(i*h),f(c)+r*c*s
in(i*h)],i=0..40)],\n           [seq([c*cos(i*h),g(c)+r*c*sin(i*h)],i=
0..40)],\n           [[c,g(c)],[c,f(c)]],[[-c,g(c)],[-c,f(c)]]],\n    \+
   linestyle=1,color=COLOR(RGB,.7,.7,.7),thickness=2): \nplots[display
]([p1,p2,p3,p4,p5,p6,p7],tickmarks=[4,4],labels=[``,``]);" }}{PARA 13 
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<I3'F87$F`ir$!3>+++_lEOfF87$Feir$!3?+++]NWkdF87$Fjir$!3#******f5z<d&F8
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^[lFa[l-F$6&7$F^[sFd_rFgjrF^[lFa[l-F$6&7$7$F]erF_[s7$F]erFg_rFgjrF^[lF
a[l-%+AXESLABELSG6%%!GFfbs-%%FONTG6#%(DEFAULTG-%*AXESTICKSG6$\"\"%F^cs
-%%VIEWG6$;F^elFcz;$F][lFh[o$\"#:Fh[o" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve
 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17
" "Curve 18" "Curve 19" "Curve 20" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x) = arctan*x;" "6#/-%\"fG6#%\"xG*
&%'arctanG\"\"\"F'F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(x) = -x/(x
^2+1);" "6#/-%\"gG6#%\"xG,$*&F'\"\"\",&*$F'\"\"#F*F*F*!\"\"F." }{TEXT 
-1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 49 "Using cylindrical shells the
 required volume is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "V[y] = 2*Pi*Int(x*(f(x)-g(x)),x = 1/sqrt(3) .. sqrt(3));" "6#/&%
\"VG6#%\"yG*(\"\"#\"\"\"%#PiGF*-%$IntG6$*&%\"xGF*,&-%\"fG6#F0F*-%\"gG6
#F0!\"\"F*/F0;*&F*F*-%%sqrtG6#\"\"$F8-F=6#F?F*" }{XPPEDIT 18 0 "`` = 2
*Pi*Int(x*(arctan*x-x/(x^2+1)),x = 1/sqrt(3) .. sqrt(3));" "6#/%!G*(\"
\"#\"\"\"%#PiGF'-%$IntG6$*&%\"xGF',&*&%'arctanGF'F-F'F'*&F-F',&*$F-F&F
'F'F'!\"\"F4F'/F-;*&F'F'-%%sqrtG6#\"\"$F4-F96#F;F'" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 2*Pi*Int(``(x*arctan*x-x^2/(x^2+1)),x = 1/sqrt(3) \+
.. sqrt(3));" "6#/%!G*(\"\"#\"\"\"%#PiGF'-%$IntG6$-F$6#,&*(%\"xGF'%'ar
ctanGF'F0F'F'*&F0F&,&*$F0F&F'F'F'!\"\"F5/F0;*&F'F'-%%sqrtG6#\"\"$F5-F:
6#F<F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 15 "This integral  " }{XPPEDIT 18 0 "Int(x*arctan*x,x)" 
"6#-%$IntG6$*(%\"xG\"\"\"%'arctanGF(F'F(F'" }{TEXT -1 57 "  can be fou
nd by using the integration by parts formula " }{XPPEDIT 18 0 "Int(u*`
`(dv/dx),x)=u*v-Int(v*``(du/dx),x)" "6#/-%$IntG6$*&%\"uG\"\"\"-%!G6#*&
%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6#*&%#duGF)F/F0
F)F1F0" }{TEXT -1 8 "  with: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "PIECEWISE([u = arctan*x, v = x^2/2],[du/dx = 1/(x^2+1),
 dv/dx = x]);" "6#-%*PIECEWISEG6$7$/%\"uG*&%'arctanG\"\"\"%\"xGF+/%\"v
G*&F,\"\"#F0!\"\"7$/*&%#duGF+%#dxGF1*&F+F+,&*$F,F0F+F+F+F1/*&%#dvGF+F6
F1F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*arctan*x,x) = x^2/2;" "6#/-%
$IntG6$*(%\"xG\"\"\"%'arctanGF)F(F)F(*&F(\"\"#F,!\"\"" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "arctan*x-1/2" "6#,&*&%'arctanG\"\"\"%\"xGF&F&*&F&F&
\"\"#!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^2/(x^2+1),x)" "6#-
%$IntG6$*&%\"xG\"\"#,&*$F'F(\"\"\"F+F+!\"\"F'" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 5 " Now " }{XPPEDIT 18 0 "Int(x^2/(x^2+1),x) \+
= Int(``(1-1/(x^2+1)),x);" "6#/-%$IntG6$*&%\"xG\"\"#,&*$F(F)\"\"\"F,F,
!\"\"F(-F%6$-%!G6#,&F,F,*&F,F,,&*$F(F)F,F,F,F-F-F(" }{XPPEDIT 18 0 "``
 = x-arctan*x+c[1];" "6#/%!G,(%\"xG\"\"\"*&%'arctanGF'F&F'!\"\"&%\"cG6
#F'F'" }{TEXT -1 64 ", and since this integral already appears in the \+
expression for " }{XPPEDIT 18 0 "V[y]" "6#&%\"VG6#%\"yG" }{TEXT -1 16 
", we note that: " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 
"Int(``(x*arctan*x-x^2/(x^2+1)),x) = x^2/2;" "6#/-%$IntG6$-%!G6#,&*(%
\"xG\"\"\"%'arctanGF-F,F-F-*&F,\"\"#,&*$F,F0F-F-F-!\"\"F3F,*&F,F0F0F3
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan*x-3/2" "6#,&*&%'arctanG\"\"\"
%\"xGF&F&*&\"\"$F&\"\"#!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^
2/(x^2+1),x);" "6#-%$IntG6$*&%\"xG\"\"#,&*$F'F(\"\"\"F+F+!\"\"F'" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` =
 x^2/2;" "6#/%!G*&%\"xG\"\"#F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "a
rctan*x-3/2*x+3/2" "6#,(*&%'arctanG\"\"\"%\"xGF&F&*(\"\"$F&\"\"#!\"\"F
'F&F+*&F)F&F*F+F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan*x+c[2]" "6#,
&*&%'arctanG\"\"\"%\"xGF&F&&%\"cG6#\"\"#F&" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "``=(x^2/2+3/2)*arctan*x-3*x/2+c[2]" "6#/%!G,(*(,&*&%\"xG\"\"#F*!
\"\"\"\"\"*&\"\"$F,F*F+F,F,%'arctanGF,F)F,F,*(F.F,F)F,F*F+F+&%\"cG6#F*
F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "V[y]=2*Pi*((x^2/2+3/2)*arct
an*x-3*x/2)" "6#/&%\"VG6#%\"yG*(\"\"#\"\"\"%#PiGF*,&*(,&*&%\"xGF)F)!\"
\"F**&\"\"$F*F)F1F*F*%'arctanGF*F0F*F**(F3F*F0F*F)F1F1F*" }{TEXT -1 1 
" " }{XPPEDIT 18 0 "PIECEWISE([sqrt(3),``],[1/sqrt(3),``])" "6#-%*PIEC
EWISEG6$7$-%%sqrtG6#\"\"$%!G7$*&\"\"\"F.-F(6#F*!\"\"F+" }{TEXT -1 1 " \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``=2*Pi*(``(3*arctan(sqrt(3))-3*sqrt(3)/2)-((1/6+3/2)*a
rctan(1/sqrt(3))-3/(2*sqrt(3))))" "6#/%!G*(\"\"#\"\"\"%#PiGF',&-F$6#,&
*&\"\"$F'-%'arctanG6#-%%sqrtG6#F.F'F'*(F.F'-F36#F.F'F&!\"\"F8F',&*&,&*
&F'F'\"\"'F8F'*&F.F'F&F8F'F'-F06#*&F'F'-F36#F.F8F'F'*&F.F'*&F&F'-F36#F
.F'F8F8F8F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 2*Pi*(``(Pi-3*sqrt(3)/
2)-(5*Pi/18-sqrt(3)/2));" "6#/%!G*(\"\"#\"\"\"%#PiGF',&-F$6#,&F(F'*(\"
\"$F'-%%sqrtG6#F.F'F&!\"\"F2F',&*(\"\"&F'F(F'\"#=F2F'*&-F06#F.F'F&F2F2
F2F'" }{XPPEDIT 18 0 "``=2*Pi*(13*Pi/18-sqrt(3))" "6#/%!G*(\"\"#\"\"\"
%#PiGF',&*(\"#8F'F(F'\"#=!\"\"F'-%%sqrtG6#\"\"$F-F'" }{TEXT -1 1 " " }
{TEXT 358 1 "~" }{TEXT -1 13 " 3.373299061 " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "f := x -> arctan(
x):\ng := x -> x/(x^2+1):\n'f(x)'=f(x);\n'g(x)'=g(x);\n2*Pi*Int(x*('f(
x)-g(x)'),x=1/sqrt(3)..sqrt(3));\n``=2*Pi*Int(x*(f(x)-g(x)),x=1/sqrt(3
)..sqrt(3));\n``=value(rhs(%));\nevalf(evalf[14](%));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%'arctanGF&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"gG6#%\"xG*&F'\"\"\",&*$)F'\"\"#F)F)F)F)!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&%\"xG
F&,&-%\"fG6#F,F&-%\"gGF0!\"\"F&/F,;,$*&\"\"$F3F8#F&F%F&*$F8F9F&F&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\"\"%#PiGF(-%$IntG6$*&%
\"xGF(,&-%'arctanG6#F.F(*&F.F(,&*$)F.F'F(F(F(F(!\"\"F7F(/F.;,$*&\"\"$F
7F<#F(F'F(*$F<F=F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#
\"\"\"%#PiGF(,&*(\"#8F(\"#=!\"\"F)F(F(*$\"\"$#F(F'F.F(F(" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/%!G$\"+h!*HtL!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}
{PARA 0 "" 0 "" {TEXT -1 89 "Find the volume of the solid of revolutio
n formed when the region bounded by the curves  " }{XPPEDIT 18 0 "y = \+
2*x;" "6#/%\"yG*&\"\"#\"\"\"%\"xGF'" }{TEXT -1 7 "  and  " }{XPPEDIT 
18 0 "y = x^2;" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 16 ", and the lines \+
" }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 28 " is rotated  (a) \+
around the " }{TEXT 304 1 "x" }{TEXT -1 22 " axis, (b) around the " }
{TEXT 359 1 "y" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT 
-1 5 "(a)  " }{XPPEDIT 18 0 "Pi*Int(``(4*x^2-x^4),x = 1 .. 2) = 47/15*
Pi;" "6#/*&%#PiG\"\"\"-%$IntG6$-%!G6#,&*&\"\"%F&*$%\"xG\"\"#F&F&*$F1F/
!\"\"/F1;F&F2F&*(\"#ZF&\"#:F4F%F&" }{TEXT -1 1 " " }{TEXT 305 1 "~" }
{TEXT -1 14 " 9.843656981  " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 5 "(b)  " }{XPPEDIT 18 0 "2*Pi*Int(x*(2*x-x^2),x = \+
1 .. 2) = 11/6*Pi" "6#/*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&%\"xGF&,&*&F%F&F
,F&F&*$F,F%!\"\"F&/F,;F&F%F&*(\"#6F&\"\"'F0F'F&" }{TEXT -1 1 " " }
{TEXT 360 1 "~" }{TEXT -1 13 " 5.759586532 " }}}{PARA 0 "" 0 "" {TEXT 
-1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________
______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 89 "Find t
he volume of the solid of revolution formed when the region bounded by
 the curves  " }{XPPEDIT 18 0 "y = 3*x-x^2;" "6#/%\"yG,&*&\"\"$\"\"\"%
\"xGF(F(*$F)\"\"#!\"\"" }{TEXT -1 6 "  and " }{XPPEDIT 18 0 "y=1/x" "6
#/%\"yG*&\"\"\"F&%\"xG!\"\"" }{TEXT -1 16 ", and the lines " }
{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 27 " is rotated (a) around the
 " }{TEXT 302 1 "x" }{TEXT -1 22 " axis, (b) around the " }{TEXT 361 
1 "y" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 
{PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a)  \+
" }{XPPEDIT 18 0 "Pi*Int(``(9*x^2-6*x^3+x^4-1/(x^2)),x = 1 .. 2) = 21/
5*Pi;" "6#/*&%#PiG\"\"\"-%$IntG6$-%!G6#,**&\"\"*F&*$%\"xG\"\"#F&F&*&\"
\"'F&*$F1\"\"$F&!\"\"*$F1\"\"%F&*&F&F&*$F1F2F7F7/F1;F&F2F&*(\"#@F&\"\"
&F7F%F&" }{TEXT -1 1 " " }{TEXT 303 1 "~" }{TEXT -1 13 " 13.19468915 \+
" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 "2*Pi*Int(x*(3*x-
x^2-1/x),x = 1 .. 2) = 9/2*Pi" "6#/*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&%\"x
GF&,(*&\"\"$F&F,F&F&*$F,F%!\"\"*&F&F&F,F1F1F&/F,;F&F%F&*(\"\"*F&F%F1F'
F&" }{TEXT -1 1 " " }{TEXT 362 1 "~" }{TEXT -1 13 " 14.13716694 " }}}
{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}
{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 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 37 "__
___________________________________" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}
{PARA 0 "" 0 "" {TEXT -1 88 "Find the volume of the solid of revolutio
n formed when the region bounded by the curve  " }{XPPEDIT 18 0 "y = x
^2+1;" "6#/%\"yG,&*$%\"xG\"\"#\"\"\"F)F)" }{TEXT -1 14 " and the line \+
" }{XPPEDIT 18 0 "y = 2;" "6#/%\"yG\"\"#" }{TEXT -1 27 " is rotated (a
) around the " }{TEXT 306 1 "x" }{TEXT -1 22 " axis, (b) around the " 
}{TEXT 363 1 "y" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT 
-1 4 "(a) " }{XPPEDIT 18 0 "Pi*Int(``(3-x^4-2*x^2),x = -1 .. 1) = 64/1
5*Pi;" "6#/*&%#PiG\"\"\"-%$IntG6$-%!G6#,(\"\"$F&*$%\"xG\"\"%!\"\"*&\"
\"#F&*$F0F4F&F2/F0;,$F&F2F&F&*(\"#kF&\"#:F2F%F&" }{TEXT -1 1 " " }
{TEXT 364 1 "~" }{TEXT -1 13 " 13.40412866 " }}{PARA 0 "" 0 "" {TEXT 
-1 4 "(b) " }{XPPEDIT 18 0 "2*Pi*Int(x*(1-x^2),x = 0 .. 1) = Pi/2" "6#
/*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&%\"xGF&,&F&F&*$F,F%!\"\"F&/F,;\"\"!F&F
&*&F'F&F%F/" }{TEXT -1 1 " " }{TEXT 365 1 "~" }{TEXT -1 14 " 1.5707963
27. " }}}{PARA 0 "" 0 "" {TEXT -1 37 "________________________________
_____" }}{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 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 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}
{PARA 0 "" 0 "" {TEXT -1 111 "Find the volume of the solid of revoluti
on formed when the region in the first quadrant bounded by the curves \+
 " }{XPPEDIT 18 0 "y = sqrt(x+1);" "6#/%\"yG-%%sqrtG6#,&%\"xG\"\"\"F*F
*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = exp(-x);" "6#/%\"yG-%$expG6
#,$%\"xG!\"\"" }{TEXT -1 14 " and the line " }{XPPEDIT 18 0 "x= 3" "6#
/%\"xG\"\"$" }{TEXT -1 28 "  is rotated (a) around the " }{TEXT 369 1 
"x" }{TEXT -1 22 " axis, (b) around the " }{TEXT 371 1 "y" }{TEXT -1 
7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" 
{TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 5 "(a)  " }{XPPEDIT 18 0 "Pi*Int(x+1-exp(-2*x),x=0..3)=Pi*(ex
p(-6)/2+7)" "6#/*&%#PiG\"\"\"-%$IntG6$,(%\"xGF&F&F&-%$expG6#,$*&\"\"#F
&F+F&!\"\"F2/F+;\"\"!\"\"$F&*&F%F&,&*&-F-6#,$\"\"'F2F&F1F2F&\"\"(F&F&
" }{TEXT -1 1 " " }{TEXT 370 1 "~" }{TEXT -1 13 " 21.99504219 " }}
{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 "2*Pi*Int(x*(sqrt(x+1
)-exp(-x)),x = 0 .. 3) = 2*Pi*(101/15+4*exp(-3))" "6#/*(\"\"#\"\"\"%#P
iGF&-%$IntG6$*&%\"xGF&,&-%%sqrtG6#,&F,F&F&F&F&-%$expG6#,$F,!\"\"F6F&/F
,;\"\"!\"\"$F&*(F%F&F'F&,&*&\"$,\"F&\"#:F6F&*&\"\"%F&-F36#,$F:F6F&F&F&
" }{TEXT -1 1 " " }{TEXT 372 1 "~" }{TEXT -1 14 "  43.55806657 " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
60 "Pi*Int(x+1-exp(-2*x),x=0..3);\nvalue(%);\nevalf(evalf[15](%));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#PiG\"\"\"-%$IntG6$,(%\"xGF%F%F%-%$
expG6#,$*&\"\"#F%F*F%!\"\"F1/F*;\"\"!\"\"$F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&%#PiG\"\"\",&*&#F%\"\"#F%-%$expG6#!\"'F%F%\"\"(F%F%" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+>U]*>#!\")" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "2*Pi*Int(x*(
sqrt(x+1)-exp(-x)),x=0..3);\nvalue(%);\nevalf(evalf[15](%));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&-%$IntG6$*&%\"xGF&,&*$
,&F,F&F&F&#F&F%F&-%$expG6#,$F,!\"\"F5F&/F,;\"\"!\"\"$F&F&" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&,&*&\"\"%F&-%$expG6#!\"$F
&F&#\"$,\"\"#:F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+dm!eN%!\")
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 37 "___________________
__________________" }}{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 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 37 "_____________________________________" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 
"Q5 " }}{PARA 0 "" 0 "" {TEXT -1 99 "Find the volume of the solid of r
evolution formed when the disk of radius 1 bounded by the curves  " }
{XPPEDIT 18 0 "y = 2+sqrt(1-x^2);" "6#/%\"yG,&\"\"#\"\"\"-%%sqrtG6#,&F
'F'*$%\"xGF&!\"\"F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 2-sqrt(1-
x^2)" "6#/%\"yG,&\"\"#\"\"\"-%%sqrtG6#,&F'F'*$%\"xGF&!\"\"F." }{TEXT 
-1 23 " is rotated around the " }{TEXT 307 1 "x" }{TEXT -1 7 " axis. \+
" }}{PARA 0 "" 0 "" {TEXT 259 5 "Notes" }{TEXT -1 2 ": " }}{PARA 15 "
" 0 "" {TEXT -1 38 "The resulting solid is a solid torus. " }}{PARA 
15 "" 0 "" {TEXT -1 26 "You may use the fact that " }{XPPEDIT 18 0 "In
t(sqrt(1-x^2),x = -1 .. 1) = Pi/2;" "6#/-%$IntG6$-%%sqrtG6#,&\"\"\"F+*
$%\"xG\"\"#!\"\"/F-;,$F+F/F+*&%#PiGF+F.F/" }{TEXT -1 63 ".  (The integ
ral gives the area of a semi-circle of radius 1.) " }}{SECT 1 {PARA 5 
"" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "8*Pi*Int(sqrt(1-x^2),x = -1 .. 1) = 4*Pi^2;" "6#/*(\"\")\"\"\"%#
PiGF&-%$IntG6$-%%sqrtG6#,&F&F&*$%\"xG\"\"#!\"\"/F0;,$F&F2F&F&*&\"\"%F&
*$F'F1F&" }{TEXT -1 1 " " }{TEXT 373 1 "~" }{TEXT -1 14 " 39.47841760.
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 228 "f := x -> 2+sqrt(1-x^2): g := x -> 2-sqrt(1-x^2):\n'
f(x)'=f(x); 'g(x)'=g(x);\nV[x]=Pi*Int('f(x)^2-g(x)'^2,x=-1..1);\n``=Pi
*Int(f(x)^2-g(x)^2,x=-1..1);\n``=Pi*Int(expand(f(x)^2-g(x)^2),x=-1..1)
;\n``=value(rhs(%));\nevalf(evalf[15](%));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"fG6#%\"xG,&\"\"#\"\"\"*$,&F*F**$)F'F)F*!\"\"#F*F)F
*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&\"\"#\"\"\"*$,&F*
F**$)F'F)F*!\"\"#F*F)F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"VG6#%
\"xG*&%#PiG\"\"\"-%$IntG6$*$),&*$)-%\"fGF&\"\"#F*F*-%\"gGF&!\"\"F5F*/F
';F8F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&%#PiG\"\"\"-%$IntG6$
,&*$),&\"\"#F'*$,&F'F'*$)%\"xGF/F'!\"\"#F'F/F'F/F'F'*$),&F/F'F0F5F/F'F
5/F4;F5F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&%#PiG\"\"\"-%$Int
G6$,$*&\"\")F',&F'F'*$)%\"xG\"\"#F'!\"\"#F'F2F'/F1;F3F'F'" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"%\"\"\")%#PiG\"\"#F(F(" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/%!G$\"+g<%y%R!\")" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 
"" {TEXT -1 37 "_____________________________________" }}{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 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 37 "____________
_________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "
;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" 
{TEXT -1 89 "Find the volume of the solid of revolution formed when th
e region bounded by the curves  " }{XPPEDIT 18 0 "y = 3-1/x;" "6#/%\"y
G,&\"\"$\"\"\"*&F'F'%\"xG!\"\"F*" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 
"y=2-1/x" "6#/%\"yG,&\"\"#\"\"\"*&F'F'%\"xG!\"\"F*" }{TEXT -1 16 ", an
d the lines " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 5 " a
nd " }{XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 27 " is rotated
 (a) around the " }{TEXT 301 1 "x" }{TEXT -1 22 " axis, (b) around the
 " }{TEXT 366 1 "y" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" 
{TEXT -1 5 "(a)  " }{XPPEDIT 18 0 "Pi*Int(``(5-2/x),x = 1 .. 3) = (10-
2*ln(3))*Pi;" "6#/*&%#PiG\"\"\"-%$IntG6$-%!G6#,&\"\"&F&*&\"\"#F&%\"xG!
\"\"F2/F1;F&\"\"$F&*&,&\"#5F&*&F0F&-%#lnG6#F5F&F2F&F%F&" }{TEXT -1 1 "
 " }{TEXT 367 1 "~" }{TEXT -1 14 " 24.51314195. " }}{PARA 0 "" 0 "" 
{TEXT -1 5 "(b)  " }{XPPEDIT 18 0 "2*Pi*Int(x,x = 1 .. 3) = 8*Pi" "6#/
*(\"\"#\"\"\"%#PiGF&-%$IntG6$%\"xG/F+;F&\"\"$F&*&\"\")F&F'F&" }{TEXT 
-1 1 " " }{TEXT 368 1 "~" }{TEXT -1 13 " 25.13274123 " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 217 "f := x -> 3-1/x: g := x -> 2-1/x:\n'f(x)'=f(x
); 'g(x)'=g(x);\nV[x]=Pi*Int('f(x)^2-g(x)'^2,x=1..3);\n``=Pi*Int(f(x)^
2-g(x)^2,x=1..3);\n``=Pi*Int(expand(f(x)^2-g(x)^2),x=1..3);\n``=value(
rhs(%));\n``=evalf(evalf[15](rhs(%)));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%\"fG6#%\"xG,&\"\"$\"\"\"*&F*F*F'!\"\"F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"gG6#%\"xG,&\"\"#\"\"\"*&F*F*F'!\"\"F," }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/&%\"VG6#%\"xG*&%#PiG\"\"\"-%$IntG6$*$),&*$)-%
\"fGF&\"\"#F*F*-%\"gGF&!\"\"F5F*/F';F*\"\"$F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%!G*&%#PiG\"\"\"-%$IntG6$,&*$),&\"\"$F'*&F'F'%\"xG!\"
\"F2\"\"#F'F'*$),&F3F'F0F2F3F'F2/F1;F'F/F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%!G*&%#PiG\"\"\"-%$IntG6$,&\"\"&F'*&\"\"#F'%\"xG!\"\"F
0/F/;F'\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&%#PiG\"\"\",&*
&\"\"#F'-%#lnG6#\"\"$F'!\"\"\"#5F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/%!G$\"+&>98X#!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 223 "f := x -> 3-1/x: g := x -> 2-1/x:\n'f(x)
'=f(x); 'g(x)'=g(x);\nV[x]=2*Pi*Int('x*(f(x)-g(x))',x=1..3);\n``=2*Pi*
Int(x*(f(x)-g(x)),x=1..3);\n``=2*Pi*Int(expand(x*(f(x)-g(x))),x=1..3);
\n``=value(rhs(%));\n``=evalf(evalf[15](rhs(%)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"fG6#%\"xG,&\"\"$\"\"\"*&F*F*F'!\"\"F," }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG,&\"\"#\"\"\"*&F*F*F'!\"\"F," }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"VG6#%\"xG,$*(\"\"#\"\"\"%#PiGF+-%
$IntG6$*&F'F+,&-%\"fGF&F+-%\"gGF&!\"\"F+/F';F+\"\"$F+F+" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\"\"%#PiGF(-%$IntG6$%\"xG/F-;F(\"
\"$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*(\"\"#\"\"\"%#PiGF(-
%$IntG6$%\"xG/F-;F(\"\"$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$
*&\"\")\"\"\"%#PiGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+BTF8
D!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 37 "___________________
__________________" }}{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 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 37 "_____________________________________" }}{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 17 "Code for \+
pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 
{PARA 0 "" 0 "" {TEXT -1 41 "vol of rev for region between two curves \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 2759 "f := x -> 2+.1*sin(x/2)+.2*cos(2*x)+x/4:\ng := x -> x^2/20+1
+.4*cos(x+1):\na := .6: b := 5.1:\nc := 3.25: d := 3.35: m := (c+d)/2:
 \nq := evalf(Pi/20): r := .13:\ndarkblue := COLOR(RGB,.5,0,.75):\np1 \+
:= plottools[arrow]([-.15,0],[6,0],0,.1,.02,arrow,color=black):\np2 :=
 plot([[[a,g(a)],[a,f(a)]],[[b,g(b)],[b,f(b)]]],color=black):\np3 := p
lot([[[a,0],[a,g(a)]],[[b,0],[b,g(b)]]],linestyle=2,color=black):\np4 \+
:= plot([[[3.3,0],[3.3,-.1]],[[2.,f(m)],[c,f(m)]],\n          [[d,g(m)
],[4.4,g(m)]],\n          [[c,g(c)],[c,3.6]],[[d,g(d)],[d,3.6]]],\n   \+
        color=darkblue,linestyle=2):\np5 := plot([f(x),g(x),-f(x),-g(x
)],x=-.15..6,color=[red,blue,red,blue],\n                 linestyle=[1
$2,2$2],thickness=[2$2,1$2]):\np6 := plot([[seq([d+r*f(m)*cos(i*q),f(m
)*sin(i*q)],i=-10..10),\n  seq([c+r*f(m)*cos(i*q),f(m)*sin(i*q)],i=10.
.30),[d,-f(m)]],\n      [seq([d+r*f(m)*cos(i*q),f(m)*sin(i*q)],i=10..3
0)],\n      [seq([c+r*g(m)*cos(i*q),g(m)*sin(i*q)],i=-10..10)],\n     \+
   [seq([d+r*g(m)*cos(i*q),g(m)*sin(i*q)],i=-10..10)],\n      [seq([d+
r*g(m)*cos(i*q),g(m)*sin(i*q)],i=10..30)]],color=darkblue):\np7 := plo
t([ [seq([c+r*f(m)*cos(i*q),f(m)*sin(i*q)],i=-10..10)],\n       [[d,g(
m)],seq([c+r*g(m)*cos(i*q),g(m)*sin(i*q)],i=10..30),[d,-g(m)]]],\n    \+
   linestyle=2,color=darkblue):\n\np8 := plot([[seq([a+r*f(a)*cos(i*q)
,f(a)*sin(i*q)],i=0..40)],\n       [seq([b+r*f(b)*cos(i*q),f(b)*sin(i*
q)],i=0..40)],\n      [seq([a+r*g(a)*cos(i*q),g(a)*sin(i*q)],i=0..40)]
,\n       [seq([b+r*g(b)*cos(i*q),g(b)*sin(i*q)],i=0..40)]],\n       l
inestyle=2,color=black):\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)):\np9 := plots[polygonplot]([
seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2..25)],\n           style
=patchnogrid,color=COLOR(RGB,.95,.95,.95)):\np10 := plottools[arrow]([
3,3.4],[c,3.4],0,.1,.3,arrow,color=darkblue):\np11 := plottools[arrow]
([3.6,3.4],[d,3.4],0,.1,.3,arrow,color=darkblue):\np12 := plottools[ar
row]([2.3,f(m)/2+.3],[2.3,f(m)],0,.09,.1,arrow,color=darkblue):\np13 :
= plottools[arrow]([2.3,f(m)/2-.3],[2.3,0],0,.1,.1,arrow,color=darkblu
e):\np14 := plottools[arrow]([4.2,g(m)/2+.3],[4.2,g(m)],0,.1,.3,arrow,
color=darkblue):\np15 := plottools[arrow]([4.2,g(m)/2-.3],[4.2,0],0,.1
,.3,arrow,color=darkblue):\nt1 := plots[textplot]([[5.2,-.18,'b'],[.5,
-.17,'a']],color=black):\nt2 := plots[textplot]([5.5,3.95,`y = f(x)`],
color=red):\nt3 := plots[textplot]([5.8,2.7,`y = g(x)`],color=blue):\n
t4 := plots[textplot]([[2.9,3.45,`x`],[3.3,-.28,`x`],\n    [2.3,f(m)/2
,`f(x)`],[4.2,g(m)/2,`g(x)`]],color=darkblue):\nt5 := plots[textplot](
[[2.8,3.45,'d']],font=[SYMBOL,10],color=darkblue):\nplots[display]([p1
,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,\n          p12,p13,p14,p15,t1,t2,t3,
t4,t5],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 57 "vol of rev for region between
 two curves by shell method " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2961 "f := x -> sin(x+1/2)-cos(2
*x+1)+x:\ng := x -> cos(x+1/2)+sin(3*x+3/2)-x:\na := .65: b := 2.3:\nc
 := 1.27: d := 1.33: m := (c+d)/2: \nq := evalf(Pi/40): r := .63:\ndar
kblue := COLOR(RGB,.5,0,.7):\np1 := plottools[arrow]([0,-3.9],[0,4.5],
0,.07,.02,arrow,color=black):\np2 := plot([[[a,g(a)],[a,f(a)]],[[b,g(b
)],[b,f(b)]]],color=black):\np3 := plot([[[-a,g(a)],[-a,f(a)]],[[-b,g(
b)],[-b,f(b)]],\n     [[a,g(a)],[a,-3.9]],[[b,g(b)],[b,-3.9]]],linesty
le=2,color=black):\np4 := plot([[[m,g(m)],[m,g(m)-.3]],[[c,f(m)],[c,f(
m)+1]],\n          [[d,f(m)],[d,f(m)+1]],[[d,f(m)],[d+.7,f(m)]],\n    \+
      [[d,g(m)],[d+.7,g(m)]]],\n           color=darkblue,linestyle=2)
:\np5 := plot([f(x),g(x)],x=.5..2.6,color=[red,blue],\n               \+
  linestyle=[1$2,2$2],thickness=2):\np6 := plot([f(-x),g(-x)],x=-2.5..
-0.5,color=[red,blue],\n                 linestyle=2,thickness=1):\n\n
p7 := plot([[seq([c*cos(i*q),f(m)+r*c*sin(i*q)],i=0..80)],\n      [seq
([d*cos(i*q),f(m)+r*d*sin(i*q)],i=0..80)],\n      [seq([d*cos(i*q),g(m
)+r*d*sin(i*q)],i=40..80)],\n      [[-d,g(m)],[-d,f(m)]],[[d,g(m)],[d,
f(m)]]],color=darkblue):\n\np8 := plot([[seq([c*cos(i*q),g(m)+r*c*sin(
i*q)],i=0..80)],\n       [seq([d*cos(i*q),g(m)+r*d*sin(i*q)],i=0..40)]
,\n       [[-c,g(m)],[-c,f(m)]],[[c,g(m)],[c,f(m)]]],\n       linestyl
e=2,color=darkblue):\n\np9 := plot([[seq([a*cos(i*q),f(a)+r*a*sin(i*q)
],i=0..80)],\n       [seq([b*cos(i*q),f(b)+r*b*sin(i*q)],i=0..80)],\n \+
     [seq([a*cos(i*q),g(a)+r*a*sin(i*q)],i=0..80)],\n       [seq([b*co
s(i*q),g(b)+r*b*sin(i*q)],i=0..80)]],\n       linestyle=2,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)):\np10 := plots[polygonplot]([seq([fpts[i-1],fpts[i]
,gpts[i],gpts[i-1]],i=2..25)],\n           style=patchnogrid,color=COL
OR(RGB,.95,.95,.95)):\n\npp := plot(f(-x),x=-b..-a,adaptive=false,nump
oints=25):\nfpts := op(1,op(1,pp)):\npp := plot(g(-x),x=-b..-a,adaptiv
e=false,numpoints=25):\ngpts := op(1,op(1,pp)):\np11 := plots[polygonp
lot]([seq([fpts[i-1],fpts[i],gpts[i],gpts[i-1]],i=2..25)],\n          \+
 style=patchnogrid,color=COLOR(RGB,.98,.98,.98)):\np12 := plottools[ar
row]([d+.4,(f(m)+g(m))/2+.3],[d+.4,f(m)],0,.05,.07,arrow,color=darkblu
e):\np13 := plottools[arrow]([d+.4,(f(m)+g(m))/2-.3],[d+.4,g(m)],0,.05
,.07,arrow,color=darkblue):\np14 := plottools[arrow]([d+.15,f(m)+.8],[
d,f(m)+.8],0,.1,.3,arrow,color=darkblue):\np15 := plottools[arrow]([c-
.15,f(m)+.8],[c,f(m)+.8],0,.1,.3,arrow,color=darkblue):\nt1 := plots[t
extplot]([[a,-4,'x = a'],[b,-4,'x = b'],[-.1,4.3,`y`]],color=black):\n
t2 := plots[textplot]([2.5,2.3,`y = f(x)`],color=red):\nt3 := plots[te
xtplot]([2.5,-2.1,`y = g(x)`],color=blue):\nt4 := plots[textplot]([[m,
-2.73,`x`],[1.33,4.45,`x`],\n    [d+.4,(f(m)+g(m))/2,`f(x) - g(x)`]],c
olor=darkblue):\nt5 := plots[textplot]([[1.28,4.45,'d']],font=[SYMBOL,
10],color=darkblue):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p
11,\n     p12,p13,p14,p15,t1,t2,t3,t4,t5],axes=none);" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 0 "" 0 "" {TEXT -1 32 " pictures of cylindrical shell  \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 2937 "r := 1.1: h := 2: rp := 1:\ndarkblue := COLOR(RGB,.5,0,.7):
\np1 := plot([r*cos(t),r*0.3*sin(t),t=-Pi..0],color=darkblue):\np2 := \+
plot([r*cos(t),r*0.3*sin(t)+h,t=0..2*Pi],color=darkblue):\nA1:=[r,0]: \+
B1:=[r,h]: C1:=[-r,h]: D1:=[-r,0]:\nAp:=[rp,0]: Bp:=[rp,h]: Cp:=[-rp,h
]: Dp:=[-rp,0]:\np3 := plot([[A1,B1],[D1,C1]],color=darkblue):\na := e
valf(Pi/60):\nc1 := seq([r*cos(j*a),r*0.3*sin(j*a)],j=-12..0):\nc2 := \+
seq([r*cos(j*a),r*0.3*sin(j*a)],j=-27..-12):\nc3 := seq([r*cos(j*a),r*
0.3*sin(j*a)],j=-33..-27):\nc4 := seq([r*cos(j*a),r*0.3*sin(j*a)],j=-4
8..-33):\nc5 := seq([r*cos(j*a),r*0.3*sin(j*a)],j=-60..-48):\nc6 := se
q([r*cos(j*a),h-r*0.3*sin(j*a)],j=0..12):\nc7 := seq([r*cos(j*a),h-r*0
.3*sin(j*a)],j=12..27):\nc8 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=27.
.33):\nc9 := seq([r*cos(j*a),h-r*0.3*sin(j*a)],j=33..48):\nc10 := seq(
[r*cos(j*a),h-r*0.3*sin(j*a)],j=48..60):\np4 := plots[polygonplot]([c1
,c6],\n    style=patchnogrid,color=COLOR(RGB,.95,.89,1)):\np5 := plots
[polygonplot]([c2,c7],\n    style=patchnogrid,color=COLOR(RGB,.95,.91,
1)):\np6 := plots[polygonplot]([c3,c8],\n    style=patchnogrid,color=C
OLOR(RGB,.953,.923,1)):\np7 := plots[polygonplot]([c4,c9],\n    style=
patchnogrid,color=COLOR(RGB,.95,.91,1)):\np8 := plots[polygonplot]([c5
,c10],\n    style=patchnogrid,color=COLOR(RGB,.95,.89,1)):\npp := seq(
[[r*cos(2*j*a),r*0.3*sin(2*j*a)+h],\n           [r*cos(2*(j+1)*a),r*0.
3*sin(2*(j+1)*a)+h],\n           [rp*cos(2*(j+1)*a),rp*0.3*sin(2*(j+1)
*a)+h],\n           [rp*cos(2*j*a),rp*0.3*sin(2*j*a)+h]],j=0..60):\np9
 := plots[polygonplot]([pp],\n    style=patchnogrid,color=COLOR(RGB,.9
53,.933,1)):\np10 := plot([[A1,[r+.25,0]],[B1,[r+.25,h]]],\n          \+
         color=darkblue,linestyle=2):\np11 := plottools[arrow]([r+.19,
h/2-.13],[r+.19,0],\n          0,.04,.1,arrow,color=darkblue):\np12 :=
 plottools[arrow]([r+.19,h/2+.13],[r+.19,h],\n          0,.04,.1,arrow
,color=darkblue):\np13 := plottools[arrow]([rp/2+.1,h],[(rp+r)/2,h],\n
          0,.07,.08,arrow,color=darkblue):\np14 := plot([[0,h],[rp/2-0
.1,h]],color=darkblue):\np15 := plot([[0,h]],style=point,symbol=cross,
color=darkblue):\np16 := plot([rp*cos(t),rp*0.3*sin(t),t=0..2*Pi],\n  \+
                                   linestyle=2,color=darkblue):\np17 :
= plot([rp*cos(t),rp*0.3*sin(t)+h,t=0..2*Pi],\n                       \+
              linestyle=1,color=darkblue):\np18 := plot([[Ap,[rp,h+0.4
]],[Dp,Cp],[B1,[r,h+0.4]],\n    [[(rp+r)/2,h-0.2],[(rp+r)/2,h+0.2]]],l
inestyle=2,color=darkblue):\np19 := plottools[arrow]([rp-.1,h+.3],[rp,
h+.3],\n          0,.06,.3,arrow,color=darkblue):\np20 := plottools[ar
row]([r+.1,h+.3],[r,h+.3],\n          0,.06,.3,arrow,color=darkblue):
\nt1 := plots[textplot]([[r+.2,h/2,`f(x) - g(x)`],[rp+.07,h+.5,'`x`'],
\n            [rp/2,h,'`x`']],font=[HELVETICA,10],color=darkblue):\nt2
 := plots[textplot]([[rp+.03,h+.5,'d']],\n                      font=[
SYMBOL,10],color=darkblue):\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p
9,\n  p10,p11,p12,p13,p14,p15,p16,p17,p18,p19,p20,t1,t2],axes=none);" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1531 "c := 7.2: h := 2:\ndarkblu
e := COLOR(RGB,.5,0,.7):\nA2 := [0,0]: B2 := [0,h]: C2 := [c,h]: D2 :=
 [c,0]:\nB3 := [.07,h+.07]: C3 := [c+.07,h+.07]: D3 := [c+.07,.07]: \n
p1 := plot([[A2,B2,C2,D2,A2],[B2,B3,C3,C2],[C3,D3,D2]],color=darkblue)
:\np2 := plots[polygonplot]([A2,B2,C2,D2],\n         style=patchnogrid
,color=COLOR(RGB,.95,.91,1)):\np3 := plots[polygonplot]([B2,B3,C3,C2],
\n         style=patchnogrid,color=COLOR(RGB,.953,.923,1)):\np4 := plo
ts[polygonplot]([C2,C3,D3,D2],\n         style=patchnogrid,color=COLOR
(RGB,.95,.89,1)):\np5 := plot([[A2,[0,-.3]],[D2,[c,-.3]],[A2,[-.7,0]],
\n        [B2,[-.7,h]],[C2,[c+.7,h]],[C3,[c+.9,h+.07]]],\n            \+
  color=darkblue,linestyle=2):\np6 := plottools[arrow]([c/2-.3,-.2],[0
,-.2],\n          0,.08,.03,arrow,color=darkblue):\np7 := plottools[ar
row]([c/2+.3,-.2],[c,-.2],\n          0,.08,.03,arrow,color=darkblue):
\np8 := plottools[arrow]([-.5,h/2+.13],[-.5,h],\n          0,.1,.07,ar
row,color=darkblue):\np9 := plottools[arrow]([-.5,h/2-.13],[-.5,0],\n \+
         0,.1,.07,arrow,color=darkblue):\np10 := plottools[arrow]([c+.
3,h-.2],[c+.5,h],\n          0,.07,.3,arrow,color=darkblue):\np11 := p
lottools[arrow]([c+.77,h+.27],[c+.57,h+.07],\n          0,.07,.3,arrow
,color=darkblue):\nt1 := plots[textplot]([[c/2,-.2,'`2    x`'],[c+.5,h
+.2,`x`],\n        [-.5,h/2,`f(x) - g(x)`]],font=[HELVETICA,10],color=
darkblue):\nt2 := plots[textplot]([[c/2+.01,-.2,`p`],[c+.42,h+.2,'d']]
,\n                            font=[SYMBOL,10],color=darkblue):\nplot
s[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,t1,t2],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 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "5 0 0" 0 
}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }