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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "Integration and area I " }}{PARA 
0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 
0 "" 0 "" {TEXT -1 18 "Version: 22.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 62 "Antiderivatives, area and the Fundamental Theorem of Calculus \+
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 83 "Suppose we want to calculate the area of the region enclo
sed between the graph of  " }{XPPEDIT 18 0 "y = x+1;" "6#/%\"yG,&%\"xG
\"\"\"F'F'" }{TEXT -1 9 " and the " }{TEXT 266 1 "x" }{TEXT -1 11 " ax
is from " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }
{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 42 "This region can be illustrated as follows." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 403 "p1 := \+
plot([x+1,[[0,0],[0,1]],[[2,0],[2,3]],[[0,1],[2,1]]],\n      x=-0.5..3
,y=0..4.2,linestyle=[1,4,4,2],\n          color=[red,black$3],thicknes
s=[2,1$3],\n             scaling=constrained,tickmarks=[4,5]):\np2 := \+
plots[polygonplot]([[0,0],[0,1],[2,1],[2,0]],color=COLOR(RGB,.5,.5,1))
:\np3 := plots[polygonplot]([[0,1],[2,3],[2,1]],color=COLOR(RGB,.5,1,.
5)):\nplots[display]([p1,p2,p3],scaling=constrained);" }}{PARA 13 "" 
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0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
356 "We can calculate the area of this region as the sum of the areas \+
of the blue rectangle and the green triangle shown to arrive at the va
lue of 4 square units.\n(Alternatively, we could use the formula for t
he area of a trapezium: area = average of length of parallel sides tim
es distance between parallel sides).\nIf the right-hand boundary of th
e region has " }{TEXT 265 1 "x" }{TEXT -1 12 " coordinate " }{XPPEDIT 
18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 8 ", where " }{TEXT 264 1 "a" }
{TEXT -1 77 " > 0, the region can be divided up in the same way to arr
ive at the formula: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "area = a+a^2/2" "6#/%%areaG,&%\"aG\"\"\"*&F&\"\"#F)!\"\"F'" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "If we replace " }{TEXT 270 1 
"a" }{TEXT -1 4 " by " }{TEXT 271 1 "x" }{TEXT -1 16 ", we obtain the \+
" }{TEXT 259 13 "area function" }{TEXT -1 9 " A where " }{XPPEDIT 18 
0 "A(x)=x+x^2/2" "6#/-%\"AG6#%\"xG,&F'\"\"\"*&F'\"\"#F+!\"\"F)" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "This area function is a
n " }{TEXT 259 15 "anti-derivative" }{TEXT -1 26 " of the original fun
ction " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 89 ", which \+
describes the sloping line giving the upper boundary line of the regio
n, that is," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`A '`(
x)=x+1" "6#/-%$A~'G6#%\"xG,&F'\"\"\"F)F)" }{XPPEDIT 18 0 "`` = f(x);" 
"6#/%!G-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "diff(x+x^2/2,x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"F%F%" }}}{EXCHG {PARA 0 "
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0 "" 0 "" {TEXT -1 57 "This remarkable fact holds in a fairly general \+
situation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 31 "Given a function f whose graph " }{XPPEDIT 18 0 "y = f(x)" "6#/
%\"yG-%\"fG6#%\"xG" }{TEXT -1 14 " is above the " }{TEXT 272 1 "x" }
{TEXT -1 102 " axis, (throughout a suitable interval at least) the fun
ction A, which gives the area under the graph " }{XPPEDIT 18 0 "y = f(
x);" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 24 " up as far as the value " }
{TEXT 273 1 "x" }{TEXT -1 17 ", has derivative " }{XPPEDIT 18 0 "`A'`(
x)" "6#-%#A'G6#%\"xG" }{TEXT -1 43 " which is the original function f,
 that is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`A '`(x)
 = f(x);" "6#/-%$A~'G6#%\"xG-%\"fG6#F'" }{TEXT -1 1 "." }}{PARA 257 "
" 0 "" {TEXT -1 2 "  " }{TEXT 262 7 "_______" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "This result is \+
the essential ingredient of the " }{TEXT 259 31 "Fundamental Theorem o
f Calculus" }{TEXT -1 39 ", which has far-reaching implications. " }}
{PARA 0 "" 0 "" {TEXT -1 98 "In order to give an explanation of this r
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18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 9 " > 0 for " }{TEXT 283 1 "x
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$$\"3')*****>C&4-OFfn$\"3w<&>i1Xv[)Ffn7$$\"35+++6j0.PFfn$\"3Il%=7-8j&)
)Ffn7$$\"3')*****z/Hez$Ffn$\"3L$H?33fT?*Ffn7$$\"3y*****R2d@!RFfn$\"30@
T3:\\T8'*Ffn7$$\"3Y*****z7es*RFfn$\"3)4.J4FO!*)**Ffn7$$\"3#******pob')
4%Ffn$\"3:s)G?#*[*R5!#;7$$\"3u******)31d>%Ffn$\"3u$ej#zu>!3\"F^`m7$$\"
3#)*************H%Ffn$\"3#***********\\C6F^`m-Fe^l6&FA$\"#5F2FTFT-%+AX
ESLABELSG6$F__lFh_l-%*AXESTICKSG6$F)F)-%%VIEWG6$;$!\"&F2F^^l;Fgam$F\\a
mF)" 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" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "A = A(x)" "6#/%\"AG-
F$6#%\"xG" }{TEXT -1 50 " be the area under the graph of f from 0 to u
p to " }{TEXT 267 1 "x" }{TEXT -1 18 ".\n\nIf we increase " }{TEXT 
268 1 "x" }{TEXT -1 6 " from " }{TEXT 269 1 "x" }{TEXT -1 4 " to " }
{XPPEDIT 18 0 "x + h" "6#,&%\"xG\"\"\"%\"hGF%" }{TEXT -1 15 ", the inc
rease " }{XPPEDIT 18 0 "Delta*A;" "6#*&%&DeltaG\"\"\"%\"AGF%" }{TEXT 
-1 14 "  in the area " }{XPPEDIT 18 0 "A(x)" "6#-%\"AG6#%\"xG" }{TEXT 
-1 122 " is greater than the area of the dark grey rectangle, but less
 than the area of the rectangle on the same base and height " }
{XPPEDIT 18 0 "f(x+h);" "6#-%\"fG6#,&%\"xG\"\"\"%\"hGF(" }{TEXT -1 50 
".\nThe dark grey rectangle with width h and height " }{XPPEDIT 18 0 "
f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 10 " has area " }{XPPEDIT 18 0 "f(x
)*`.`*h;" "6#*(-%\"fG6#%\"xG\"\"\"%\".GF(%\"hGF(" }{TEXT -1 58 ", and \+
the larger rectangle with the same base and height  " }{XPPEDIT 18 0 "
f(x+h);" "6#-%\"fG6#,&%\"xG\"\"\"%\"hGF(" }{TEXT -1 10 " has area " }
{XPPEDIT 18 0 "f(x+h)*`.`*h;" "6#*(-%\"fG6#,&%\"xG\"\"\"%\"hGF)F)%\".G
F)F*F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "Writing " }
{XPPEDIT 18 0 "Delta*x;" "6#*&%&DeltaG\"\"\"%\"xGF%" }{TEXT -1 6 "  fo
r " }{TEXT 275 1 "h" }{TEXT -1 36 " we see that the change in the area
 " }{XPPEDIT 18 0 "Delta*A;" "6#*&%&DeltaG\"\"\"%\"AGF%" }{TEXT -1 13 
" is between  " }{XPPEDIT 18 0 "f(x)*`.`*Delta*x;" "6#**-%\"fG6#%\"xG
\"\"\"%\".GF(%&DeltaGF(F'F(" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "f(x
+Delta*x)*`.`*Delta*x;" "6#**-%\"fG6#,&%\"xG\"\"\"*&%&DeltaGF)F(F)F)F)
%\".GF)%&DeltaGF)F(F)" }{TEXT -1 10 ", that is:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "f(x)
*`.`*Delta*x <= Delta*A;" "6#1**-%\"fG6#%\"xG\"\"\"%\".GF)%&DeltaGF)F(
F)*&%&DeltaGF)%\"AGF)" }{XPPEDIT 18 0 "`` <= f(x+Delta*x)*`.`*Delta*x;
" "6#1%!G**-%\"fG6#,&%\"xG\"\"\"*&%&DeltaGF+F*F+F+F+%\".GF+%&DeltaGF+F
*F+" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 276 
19 "___________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
37 "Dividing this (double) inequality by " }{XPPEDIT 18 0 "Delta*x;" "
6#*&%&DeltaG\"\"\"%\"xGF%" }{TEXT -1 12 " we obtain: " }}{PARA 257 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) <= Delta*A/(Delta*x);" "6#1-%
\"fG6#%\"xG*(%&DeltaG\"\"\"%\"AGF**&%&DeltaGF*F'F*!\"\"" }{XPPEDIT 18 
0 "`` <= f(x+Delta*x);" "6#1%!G-%\"fG6#,&%\"xG\"\"\"*&%&DeltaGF*F)F*F*
" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 277 19 "_
__________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "tha
t is,  " }{XPPEDIT 18 0 "Delta*A/(Delta*x);" "6#*(%&DeltaG\"\"\"%\"AGF
%*&%&DeltaGF%%\"xGF%!\"\"" }{TEXT -1 15 "  lies between " }{XPPEDIT 
18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 "f(
x+Delta*x);" "6#-%\"fG6#,&%\"xG\"\"\"*&%&DeltaGF(F'F(F(" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "Delta*x;" 
"6#*&%&DeltaG\"\"\"%\"xGF%" }{TEXT -1 15 " tend to zero, " }{XPPEDIT 
18 0 "f(x+Delta*x);" "6#-%\"fG6#,&%\"xG\"\"\"*&%&DeltaGF(F'F(F(" }
{TEXT -1 10 " tends to " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }
{TEXT -1 33 " (provided f is continuous) with " }{XPPEDIT 18 0 "Delta*
A/(Delta*x);" "6#*(%&DeltaG\"\"\"%\"AGF%*&%&DeltaGF%%\"xGF%!\"\"" }
{TEXT -1 21 "  sandwiched between " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#
%\"xG" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 "f(x+Delta*x);" "6#-%\"fG6#
,&%\"xG\"\"\"*&%&DeltaGF(F'F(F(" }{TEXT -1 19 " for any choice of " }
{XPPEDIT 18 0 "Delta*x;" "6#*&%&DeltaG\"\"\"%\"xGF%" }{TEXT -1 2 ". " 
}}{PARA 0 "" 0 "" {TEXT -1 53 "We can therefore make two deductions si
multaneously: " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d*A
/(d*x) = limit(Delta*A/(Delta*x),Delta*x = 0);" "6#/*(%\"dG\"\"\"%\"AG
F&*&F%F&%\"xGF&!\"\"-%&limitG6$*(%&DeltaGF&%\"AGF&*&F/F&%\"xGF&F*/*&%&
DeltaGF&F)F&\"\"!" }{TEXT -1 10 "  exists, " }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{TEXT 278 15 "_______________" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "d*A/(d*x) = f(x);" "6#/*(%\"dG\"\"\"%\"AGF&*&F%F&%\"xGF
&!\"\"-%\"fG6#F)" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 1 " " 
}{TEXT 279 7 "_______" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 60 "Strictly speaking, we should look at \+
a second picture where " }{XPPEDIT 18 0 "Delta*x = h;" "6#/*&%&DeltaG
\"\"\"%\"xGF&%\"hG" }{TEXT -1 85 " is negative, but it should be reaso
nably clear as to what will happen in that case.\n" }}{PARA 0 "" 0 "" 
{TEXT -1 232 "The Fundamental Theorem of Calculus applies to decreasin
g functions and to functions which may be both increasing and decreasi
ng in a given interval. With a suitable interpretation, it can also be
 used when the graph goes below the " }{TEXT 274 1 "x" }{TEXT -1 7 " a
xis. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 7 "
Example" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 77 "We Fundamenta
l Theorem of Calculus to calculate the area under the graph of  " }
{XPPEDIT 18 0 "y = x^2/2+2;" "6#/%\"yG,&*&%\"xG\"\"#F(!\"\"\"\"\"F(F*
" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "x = 1" "6#/%\"xG\"\"\"" }{TEXT 
-1 4 " to " }{XPPEDIT 18 0 "x = 2" "6#/%\"xG\"\"#" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
382 "f := x -> x^2/2+2:\np1 := plot(f(x),x=0..2.2,y=0..4.5,color=red,t
ickmarks=[3,5]):\ncrv := op(op(1,op(1,plot(f(x),x=1..2)))):\np2 := plo
ts[polygonplot]([[1.5,0],[1,0],crv,[2,0]],\n          style=patchnogri
d,color=COLOR(RGB,.8,.8,.8)):\np3 := plot([[[1,0],[1,f(1)]],[[2,0],[2,
f(2)]]],\n                         color=black):\nt1 := textplot([1.5,
1.5,`area`]):\nplots[display]([p1,p2,p3,t1]);" }}{PARA 13 "" 1 "" 
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1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 70 "We want to find the area of the grey \+
shaded region.\nThe area function " }{XPPEDIT 18 0 "A(x)" "6#-%\"AG6#%
\"xG" }{TEXT -1 57 " which measures the area under the graph starting \+
at the " }{TEXT 280 1 "y" }{TEXT -1 30 " axis has derivative given by \+
" }{XPPEDIT 18 0 "f(x) = x^2/2+2;" "6#/-%\"fG6#%\"xG,&*&F'\"\"#F*!\"\"
\"\"\"F*F," }{TEXT -1 37 ". \nIt must therefore be of the form  " }
{XPPEDIT 18 0 "A(x) = x^3/6+2*x+c;" "6#/-%\"AG6#%\"xG,(*&F'\"\"$\"\"'!
\"\"\"\"\"*&\"\"#F-F'F-F-%\"cGF-" }{TEXT -1 9 ",  where " }{TEXT 263 
1 "c" }{TEXT -1 27 " is an arbitrary constant. " }}{PARA 0 "" 0 "" 
{TEXT -1 11 "This is an " }{TEXT 259 15 "anti-derivative" }{TEXT -1 7 
" or an " }{TEXT 259 19 "indefinite integral" }{TEXT 258 1 " " }{TEXT 
-1 3 "of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "int(x^2/2+2,x) = x^3/
6+2*x+c;" "6#/-%$intG6$,&*&%\"xG\"\"#F*!\"\"\"\"\"F*F,F),(*&F)\"\"$\"
\"'F+F,*&F*F,F)F,F,%\"cGF," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 50 "So that the area starts off with the value 0 when
 " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 23 ", we need the co
nstant " }{TEXT 281 1 "c" }{TEXT -1 16 " to be 0. Hence " }{XPPEDIT 
18 0 "A(x) = x^3/6+2*x" "6#/-%\"AG6#%\"xG,&*&F'\"\"$\"\"'!\"\"\"\"\"*&
\"\"#F-F'F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The are
a from " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }
{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 5 " is  " }{XPPEDIT 18 
0 "A(1) = 13/6;" "6#/-%\"AG6#\"\"\"*&\"#8F'\"\"'!\"\"" }{TEXT -1 24 " \+
 and the area from the " }{TEXT 282 1 "y" }{TEXT -1 9 " axis to " }
{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 5 " is  " }{XPPEDIT 18 
0 "A(2) = 16/3;" "6#/-%\"AG6#\"\"#*&\"#;\"\"\"\"\"$!\"\"" }{TEXT -1 
32 ".\nThe area under the graph from " }{XPPEDIT 18 0 "x=1" "6#/%\"xG
\"\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }
{TEXT -1 4 " is " }{XPPEDIT 18 0 "A(2)-A(1) = 19/6;" "6#/,&-%\"AG6#\"
\"#\"\"\"-F&6#F)!\"\"*&\"#>F)\"\"'F," }{TEXT -1 8 "  or  3 " }
{XPPEDIT 18 0 "1/6;" "6#*&\"\"\"F$\"\"'!\"\"" }{TEXT -1 87 ".\nNote th
at, as a rough check on the value just obtained, a trapezium with vert
ices at " }{XPPEDIT 18 0 "``(1,0), ``(1,f(1), ``(2,f(2))" "6$-%!G6$\"
\"\"\"\"!-F$6%F&-%\"fG6#F&-F$6$\"\"#-F+6#F/" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "``(2,0)" "6#-%!G6$\"\"#\"\"!" }{TEXT -1 12 " has area 3
 " }{XPPEDIT 18 0 "1/4;" "6#*&\"\"\"F$\"\"%!\"\"" }{TEXT -1 156 ", whi
ch is slightly greater than the area under the curve. This makes sense
 because the curve goes a little bit below the sloping side of such a \+
trapezium. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 331 "f := x -> x^2/2+2:\np1 := plot(f(x),x=0..2.2,y=0..4.
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):\np2 := plots[polygonplot]([[1.5,0],[1,0],crv,[2,0]],\n         colo
r=COLOR(RGB,.7,.7,.7),style=patchnogrid):\np3 := plot([[1,0],[1,f(1)],
[2,f(2)],[2,0],[1,0]],color=blue,thickness=2):\nplots[display]([p1,p2,
p3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 278 390 390 {PLOTDATA 2 "6(-%'CURV
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0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2
" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "We can get the area function a
s an indefinite integral. Note that Maple does not give an arbitrary c
onstant." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "Int(x^2/2+2,x);\nvalue(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$,&*$)%\"xG\"\"#\"\"\"#F+F*F*F+F)" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"$\"\"\"#F(\"\"'*&\"\"#F(F&F(F(" }}
}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 144 "The ar
ea under the graph is the definite integral, which is the same as eval
uating the indefinite integral at 2 and subtracting the value at 1. " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 41 "A := x->x^3/6+2*x;\nA(2);\nA(1);\nA(2)-A(1);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"AGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"$
\"\"\"#F1\"\"'*&\"\"#F1F/F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
##\"#;\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#8\"\"'" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6##\"#>\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 124 "We introduce notation to handle the situation which aris
es in the calculation of areas under graphs of functions and write: " 
}}{PARA 257 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "Int(f(x),x = a .. \+
b) = F(x);" "6#/-%$IntG6$-%\"fG6#%\"xG/F*;%\"aG%\"bG-%\"FG6#F*" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([b, ``],[a, ``]);" "6#-%*PIEC
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{TEXT -1 2 "  " }{XPPEDIT 18 0 " ``= F(b)-F(a)" "6#/%!G,&-%\"FG6#%\"bG
\"\"\"-F'6#%\"aG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "
where " }{XPPEDIT 18 0 "`F '`(x) = f(x);" "6#/-%$F~'G6#%\"xG-%\"fG6#F'
" }{TEXT -1 17 ", that is, where " }{XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%
\"xG" }{TEXT -1 26 " is an anti-derivative of " }{XPPEDIT 18 0 "f(x)" 
"6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(f(x),x = a .. b)" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"a
G%\"bG" }{TEXT -1 15 " is called the " }{TEXT 259 17 "definite integra
l" }{TEXT -1 18 " of the function  " }{XPPEDIT 18 0 "f(x)" "6#-%\"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 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x)" "6#-%\"FG6#
%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([b, ``],[a, ``]);" "6
#-%*PIECEWISEG6$7$%\"bG%!G7$%\"aGF(" }{TEXT -1 50 "provides a signal t
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G" }{TEXT -1 4 " at " }{TEXT 297 1 "b" }{TEXT -1 28 ", and subtract th
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{PARA 0 "" 0 "" {TEXT -1 16 "If the graph of " }{XPPEDIT 18 0 "f(x)" "
6#-%\"fG6#%\"xG" }{TEXT -1 14 " is above the " }{TEXT 285 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 99 ", then this definite integral gives the area of the regio
n enclosed between the graph of the curve " }{XPPEDIT 18 0 "y=f(x)" "6
#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 6 ", the " }{TEXT 288 1 "x" }{TEXT -1 
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1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Cur
ve 9" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 54 "In many cases the \"obvious choice\" of anti-derivat
ive " }{XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\"xG" }{TEXT -1 74 ", obtained
 by omitting the usual arbitrary constant, has the value 0 when " }
{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 15 ". In this case " }
{XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\"xG" }{TEXT -1 76 " can be interpret
ed as the function which gives the area under the graph of " }
{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 46 " (that is, the ar
ea between the graph and the " }{TEXT 289 1 "x" }{TEXT -1 16 " axis) f
rom the " }{TEXT 292 1 "y" }{TEXT -1 42 " axis as far as the vertical \+
line through " }{TEXT 290 1 "x" }{TEXT -1 8 " on the " }{TEXT 291 1 "x
" }{TEXT -1 6 " axis." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 
18 0 "F(a)" "6#-%\"FG6#%\"aG" }{TEXT -1 41 " gives the area under the \+
graph from the " }{TEXT 293 1 "y" }{TEXT -1 9 " axis to " }{XPPEDIT 
18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 8 ", while " }{XPPEDIT 18 0 "F(b)
" "6#-%\"FG6#%\"bG" }{TEXT -1 41 " gives the area under the graph from
 the " }{TEXT 294 1 "y" }{TEXT -1 9 " axis to " }{XPPEDIT 18 0 "x=b" "
6#/%\"xG%\"bG" }{TEXT -1 6 ". If  " }{XPPEDIT 18 0 "a<b" "6#2%\"aG%\"b
G" }{TEXT -1 33 ", then it is clear why computing " }{XPPEDIT 18 0 "F(
b)-F(a)" "6#,&-%\"FG6#%\"bG\"\"\"-F%6#%\"aG!\"\"" }{TEXT -1 37 " gives
 the area under the graph 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 3 "If " }{XPPEDIT 18 0 "F(0)<>0" "6#0-%
\"FG6#\"\"!F'" }{TEXT -1 32 ", we could determine a constant " }{TEXT 
299 1 "c" }{TEXT -1 24 " such that the function " }{XPPEDIT 18 0 "G(x)
= F(x) +c" "6#/-%\"GG6#%\"xG,&-%\"FG6#F'\"\"\"%\"cGF," }{TEXT -1 11 " \+
satisfies " }{XPPEDIT 18 0 "G(0)=0" "6#/-%\"GG6#\"\"!F'" }{TEXT -1 14 
", and replace " }{XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\"xG" }{TEXT -1 4 "
 by " }{XPPEDIT 18 0 "G(x)" "6#-%\"GG6#%\"xG" }{TEXT -1 50 ". However,
 it is not necessary to do this because " }{XPPEDIT 18 0 "G(b)-G(a) = \+
``(F(b)+c)-(F(a)+c);" "6#/,&-%\"GG6#%\"bG\"\"\"-F&6#%\"aG!\"\",&-%!G6#
,&-%\"FG6#F(F)%\"cGF)F),&-F46#F,F)F6F)F-" }{XPPEDIT 18 0 "``=F(b)-F(a)
" "6#/%!G,&-%\"FG6#%\"bG\"\"\"-F'6#%\"aG!\"\"" }{TEXT -1 15 ", that is
, the " }{TEXT 300 1 "c" }{TEXT -1 17 "'s \"cancel out\". " }}{PARA 0 
"" 0 "" {TEXT -1 15 "Alternatively, " }{XPPEDIT 18 0 "F(0) <> 0" "6#0-
%\"FG6#\"\"!F'" }{TEXT -1 12 " means that " }{XPPEDIT 18 0 "F(x)" "6#-
%\"FG6#%\"xG" }{TEXT -1 63 " measures areas under the graph starting a
t some vertical line " }{XPPEDIT 18 0 "x=d" "6#/%\"xG%\"dG" }{TEXT -1 
29 " which is different from the " }{TEXT 298 1 "y" }{TEXT -1 42 " axi
s ( as long as we can find some value " }{XPPEDIT 18 0 "x=d" "6#/%\"xG
%\"dG" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "F(d)=0" "6#/-%\"FG6#
%\"dG\"\"!" }{TEXT -1 4 " ). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 12 "The numbers " }{TEXT 286 1 "a" }{TEXT -1 
5 " and " }{TEXT 287 1 "b" }{TEXT -1 125 ", which appear below and abo
ve the integral sign, indicate the vertical boundaries of the region i
nvolved and are called the " }{TEXT 259 22 "lower and upper limits" }
{TEXT -1 42 " of the (definite) integral respectively. " }}{PARA 0 "" 
0 "" {TEXT -1 41 "For example, the area between the curve  " }
{XPPEDIT 18 0 "y=x^2/2+2" "6#/%\"yG,&*&%\"xG\"\"#F(!\"\"\"\"\"F(F*" }
{TEXT -1 9 " and the " }{TEXT 295 1 "x" }{TEXT -1 11 " axis from " }
{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 4 " to " }{XPPEDIT 18 
0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 32 " can be calculated as follows.  \+
" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^2/2+2,x=1..
2)" "6#-%$IntG6$,&*&%\"xG\"\"#F)!\"\"\"\"\"F)F+/F(;F+F)" }{TEXT -1 1 "
 " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x^3/6+2*x;
" "6#/%!G,&*&%\"xG\"\"$\"\"'!\"\"\"\"\"*&\"\"#F+F'F+F+" }{TEXT -1 3 " \+
  " }{XPPEDIT 18 0 "PIECEWISE([2, ``],[1, ``]);" "6#-%*PIECEWISEG6$7$
\"\"#%!G7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= (8/6+4)-(1/6+2)" 
"6#/%!G,(*&\"\")\"\"\"\"\"'!\"\"F(\"\"%F(,&*&F(F(F)F*F(\"\"#F(F*" }
{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= \+
19/6" "6#/%!G*&\"#>\"\"\"\"\"'!\"\"" }{XPPEDIT 18 0 "``= 3" "6#/%!G\"
\"$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/6" "6#*&\"\"\"F$\"\"'!\"\"" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 50 "In Maple definite integrals are obtained by using " }
{TEXT 0 3 "int" }{TEXT -1 4 " or " }{TEXT 0 3 "Int" }{TEXT -1 38 " wit
h the range of values to be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Int(x^2/2+2,x=1..2);\nvalue(
%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*$)%\"xG\"\"#\"\"\"#
F+F*F*F+/F);F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#>\"\"'" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 85 "Examples of calculating
 areas under graphs using the Fundamental Theorem of Calculus " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";
" }}}{PARA 0 "" 0 "" {TEXT 305 8 "Question" }{TEXT -1 2 ": " }}{PARA 
0 "" 0 "" {TEXT -1 49 "Find the area of the region bounded by the curv
e " }{XPPEDIT 18 0 "y=x^2-4*x+5" "6#/%\"yG,(*$%\"xG\"\"#\"\"\"*&\"\"%F
)F'F)!\"\"\"\"&F)" }{TEXT -1 6 ", the " }{TEXT 304 1 "x" }{TEXT -1 29 
" axis and the vertical lines " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" 
}{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=4" "6#/%\"xG\"\"%" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT 306 8 "Solution" }{TEXT -1 2 ": " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "f
 := x -> x^2-4*x+5:\np1 := plot(f(x),x=0..4.2,y=0..5.5,color=red,thick
ness=2):\na := 1: b := 4:\np2 := plot(f(x),x=a..b,color=grey,filled=tr
ue):\np3 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b)]]],color=black):\npl
ots[display]([p1,p2,p3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 403 334 334 
{PLOTDATA 2 "6(-%'CURVESG6%7S7$$\"\"!F)$\"\"&F)7$$\"3s+++]Z![:*!#>$\"3
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3LA/GV3([-%F27$$\"3v++]27f4NF6$\"358wV@eL>PF27$$\"3B++D'R_qS%F6$\"3GGH
t\\,SJMF27$$\"3[++D,Q6R_F6$\"3/#[CPhP)yJF27$$\"3%)***\\7>l15'F6$\"3F%H
d73:>$HF27$$\"3?,+DrGo\"*pF6$\"3%fI@_9j@p#F27$$\"3?++DYI%)zyF6$\"3'[MN
e/#)*oCF27$$\"3)4++]&\\U$z)F6$\"3-n@RUK(eD#F27$$\"3U,+](R3\")f*F6$\"3'
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\"3u**p$Hz4!y9F27$$\"31++]\"yqKS\"F2$\"3)4i6-w&3c8F27$$\"3;+++YXX$[\"F
2$\"33m![g?>oE\"F27$$\"35+]7Etsw:F2$\"3%>H\\kvf\"z6F27$$\"39+++T&*Gf;F
2$\"3a!*>(oh$3;6F27$$\"3G+]7wL()\\<F2$\"3w06(zKjD1\"F27$$\"3>+]P318O=F
2$\"3qZ:^xJ&o-\"F27$$\"3E++D\")48E>F2$\"3.)3J>jca+\"F27$$\"3`+]i=%z(3?
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2$\"3&=D<^?+j.\"F27$$\"3/+](e+M6F#F2$\"3!\\#=9\\O^t5F27$$\"3=++DBF>eBF
2$\"3U^#*)p--$G6F27$$\"3E+++(*G8[CF2$\"3f7OP$4B3?\"F27$$\"30++D5=7ODF2
$\"37PdU&fEuG\"F27$$\"3D+]73cD@EF2$\"3WjniI&efQ\"F27$$\"3D++vv@y:FF2$
\"3K/97BTM7:F27$$\"3(******4]=2!GF2$\"3O(oVy6]6k\"F27$$\"3]++]dhS\"*GF
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$\"3VDs)z/p38#F27$$\"3!***\\7'Gcz9$F2$\"3S-R&[j.yJ#F27$$\"3K+]iYwJOKF2
$\"3@zjNB8[GDF27$$\"3O++]FqqALF2$\"3w')yf!)Qb\\FF27$$\"32+]7.([JT$F2$
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2$\"3Y*)[Ib/)f_$F27$$\"3U+]7sUnxOF2$\"3p],Nj4f9QF27$$\"3c+++</&)ePF2$
\"3Iv5Q*yaN4%F27$$\"3[++vRu)=&QF2$\"3J7Q]*3([HWF27$$\"3`+++i35NRF2$\"3
%[98hM:Yu%F27$$\"3G+]7EP#Q-%F2$\"3))ykUuC'e4&F27$$\"3*3+vy#Gu3TF2$\"3Y
bU69jzYaF27$$\"3;+++++++UF2$\"3&)************ReF2-%'COLOURG6&%$RGBG$\"
*++++\"!\")F(F(-%*THICKNESSG6#\"\"#-%)POLYGONSG6U7&7$$\"\"\"F)F)7$Fi[l
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$\"3sN70`!\\@m\"F27$F\\]lF)7&F`]lF[]l7$$\"33++D'3&o]7F2$\"3))p\"o*RGZh
:F27$Fc]lF)7&Fg]lFb]l7$$\"3/+](oX*y98F2$\"3od'4%)[8&p9F27$Fj]lF)7&F^^l
Fi]l7$$\"3#***\\P9CAu8F2$\"3Ca1n'e(f\"R\"F27$Fa^lF)7&Fe^lF`^l7$$\"3.+]
P*zhdV\"F2$\"3mPX/ZZO=8F27$Fh^lF)7&F\\_lFg^l7$$\"3++]P>fS*\\\"F2$\"37*
ycNV%f]7F27$F__lF)7&Fc_lF^_l7$$\"35+](=$f%Gc\"F2$\"3AP#y#zO5\">\"F27$F
f_lF)7&Fj_lFe_l7$$\"3$*****\\#y,\"G;F2$\"3Wx'zTG3$Q6F27$F]`lF)7&Fa`lF
\\`l7$$\"37++Dr\"zbo\"F2$\"3eO=bd/'))4\"F27$Fd`lF)7&Fh`lFc`l7$$\"3%***
**\\(4&G]<F2$\"3_%eID`dB1\"F27$F[alF)7&F_alFj`l7$$\"34++]7nD:=F2$\"3w2
jF#3IT.\"F27$FbalF)7&FfalFaal7$$\"3%*****\\-*oy(=F2$\"3lal(\\+;\\,\"F2
7$FialF)7&F]blFhal7$$\"35+]PpnsM>F2$\"3jCij%fgU+\"F27$F`blF)7&FdblF_bl
7$$\"3,++]siL-?F2$\"3&R>p\"ea++5F27$FgblF)7&F[clFfbl7$$\"3-+++!R5'f?F2
$\"3%>&ff)R`N+\"F27$F^clF)7&FbclF]cl7$$\"3)***\\P/QBE@F2$\"3z&)pOt\\$f
,\"F27$FeclF)7&FiclFdcl7$$\"3!******\\\"o?&=#F2$\"3-WCKk:IM5F27$F\\dlF
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7$$\"3&)**\\7j=_6BF2$\"3Sr[?re/(4\"F27$FjdlF)7&F^elFidl7$$\"33++vVy!eP
#F2$\"3cCSUN:BT6F27$FaelF)7&FeelF`el7$$\"3K+](=WU[V#F2$\"3G8eD\\z3*=\"
F27$FhelF)7&F\\flFgel7$$\"3)****\\7B>&)\\#F2$\"3!44ERU@&[7F27$F_flF)7&
FcflF^fl7$$\"3)***\\P>:mkDF2$\"3$Q)GYJE%)=8F27$FfflF)7&FjflFefl7$$\"3'
***\\iv&QAi#F2$\"3sGe*\\%3=(Q\"F27$F]glF)7&FaglF\\gl7$$\"31++vtLU%o#F2
$\"39K``a`Vo9F27$FdglF)7&FhglFcgl7$$\"37+++bjm[FF2$\"3/')*36J,0c\"F27$
F[hlF)7&F_hlFjgl7$$\"3\"****\\(yb^6GF2$\"3%p%RbMvbe;F27$FbhlF)7&FfhlFa
hl7$$\"3)***\\PMaKsGF2$\"3%G`dMm^4w\"F27$FihlF)7&F]ilFhhl7$$\"3=++D6W%
)RHF2$\"3%*fyN<vI$)=F27$F`ilF)7&FdilF_il7$$\"3z*****\\@80+$F2$\"3,N'*Q
$pE5+#F27$FgilF)7&F[jlFfil7$$\"31++]7,HlIF2$\"3Di-zBI%[8#F27$F^jlF)7&F
bjlF]jl7$$\"3()**\\P4w)R7$F2$\"3uE&Gg9[LE#F27$FejlF)7&FijlFdjl7$$\"3;+
+]x%f\")=$F2$\"3>tI(R%Hs6CF27$F\\[mF)7&F`[mF[[m7$$\"3!)**\\P/-a[KF2$\"
37n2%>k_)eDF27$Fc[mF)7&Fg[mFb[m7$$\"3/+](=Yb;J$F2$\"3%)>lm]+W?FF27$Fj[
mF)7&F^\\mFi[m7$$\"3')****\\i@OtLF2$\"3mvmQHO7')GF27$Fa\\mF)7&Fe\\mF`
\\m7$$\"3')**\\PfL'zV$F2$\"3_M]!HiQx1$F27$Fh\\mF)7&F\\]mFg\\m7$$\"3>++
+!*>=+NF2$\"3!3O?J+Y0D$F27$F_]mF)7&Fc]mF^]m7$$\"3-++DE&4Qc$F2$\"3K]-RM
-]XMF27$Ff]mF)7&Fj]mFe]m7$$\"3=+]P%>5pi$F2$\"3M,8c!yOok$F27$F]^mF)7&Fa
^mF\\^m7$$\"39+++bJ*[o$F2$\"3'R&ewV\\')QQF27$Fd^mF)7&Fh^mFc^m7$$\"33++
Dr\"[8v$F2$\"37>2%pT?s1%F27$F[_mF)7&F__mFj^m7$$\"3++++Ijy5QF2$\"3%*o[
\"H8Z*yUF27$Fb_mF)7&Ff_mFa_m7$$\"31+]P/)fT(QF2$\"3))Q\\Ls\\Z7XF27$Fi_m
F)7&F]`mFh_m7$$\"31+]i0j\"[$RF2$\"33uB^OT^VZF27$F``mF)7&Fd`mF_`m7$$\"
\"%F)F*7$Fg`mF)7\"-%&STYLEG6#%,PATCHNOGRIDG-Fjz6&F\\[l$\")=THvF_[lFaam
Faam-F$6$7$7$Fi[lF(F[\\l-Fjz6&F\\[lF)F)F)-F$6$7$7$Fg`mF(Ff`mFgam-%+AXE
SLABELSG6%Q\"x6\"Q\"yFabm-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"#U!\"\";F
($\"#bF]cm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C
urve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 52 "The required area is given by the definite integral " }
{XPPEDIT 18 0 "Int(``(x^2-4*x+5),x = 1 .. 4);" "6#-%$IntG6$-%!G6#,(*$%
\"xG\"\"#\"\"\"*&\"\"%F-F+F-!\"\"\"\"&F-/F+;F-F/" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 24 "We can use the function " }{XPPEDIT 18 0 
"F(x) = x^3/3-2*x^2+5*x;" "6#/-%\"FG6#%\"xG,(*&F'\"\"$F*!\"\"\"\"\"*&
\"\"#F,*$F'F.F,F+*&\"\"&F,F'F,F," }{TEXT -1 26 " as an antiderivative \+
for " }{XPPEDIT 18 0 "f(x)=x^2-4*x+5" "6#/-%\"fG6#%\"xG,(*$F'\"\"#\"\"
\"*&\"\"%F+F'F+!\"\"\"\"&F+" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "F(x)
" "6#-%\"FG6#%\"xG" }{TEXT -1 35 " gives the area under the graph of \+
" }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 10 " from the " }
{TEXT 307 1 "y" }{TEXT -1 42 " axis as far as the vertical line throug
h " }{TEXT 308 1 "x" }{TEXT -1 8 " on the " }{TEXT 309 1 "x" }{TEXT 
-1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 12 "In detail:  " }}{PARA 
257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(``(x^2-4*x+5),x = 1 .. \+
4) = x^3/3-2*x^2+5*x;" "6#/-%$IntG6$-%!G6#,(*$%\"xG\"\"#\"\"\"*&\"\"%F
.F,F.!\"\"\"\"&F./F,;F.F0,(*&F,\"\"$F7F1F.*&F-F.*$F,F-F.F1*&F2F.F,F.F.
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([4, ``],[``, ``],[1, ``]);
" "6#-%*PIECEWISEG6%7$\"\"%%!G7$F(F(7$\"\"\"F(" }{TEXT -1 1 " " }}
{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(64/3-32+20)-(
1/3-2+5);" "6#/%!G,&-F$6#,(*&\"#k\"\"\"\"\"$!\"\"F+\"#KF-\"#?F+F+,(*&F
+F+F,F-F+\"\"#F-\"\"&F+F-" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "``=6" "6#/%!G\"\"'" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "In
t(x^2-4*x+5,x=1..4);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%$IntG6$,(*$)%\"xG\"\"#\"\"\"F+*&\"\"%F+F)F+!\"\"\"\"&F+/F);F+F-" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"'" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT 311 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 49 
"Find the area of the region bounded by the curve " }{XPPEDIT 18 0 "y \+
= 1+2*x^2-x^3/2;" "6#/%\"yG,(\"\"\"F&*&\"\"#F&*$%\"xGF(F&F&*&F*\"\"$F(
!\"\"F-" }{TEXT -1 6 ", the " }{TEXT 310 1 "x" }{TEXT -1 29 " axis and
 the vertical lines " }{XPPEDIT 18 0 "x = -1;" "6#/%\"xG,$\"\"\"!\"\"
" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT 
-1 2 ". " }}{PARA 0 "" 0 "" {TEXT 312 8 "Solution" }{TEXT -1 2 ": " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
239 "f := x -> 1+2*x^2-x^3/2:\np1 := plot(f(x),x=-1.2..4.2,y=-.5..6,co
lor=red,thickness=2):\na := -1: b := 3:\np2 := plot(f(x),x=a..b,color=
grey,filled=true):\np3 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b)]]],col
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"Curve 1" "Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 22 "The required area is: " }}{PARA 257 "" 0 "" {TEXT -1 2 " \+
 " }{XPPEDIT 18 0 "Int(``(1+2*x^2-x^3/2),x = -1 .. 3);" "6#-%$IntG6$-%
!G6#,(\"\"\"F**&\"\"#F**$%\"xGF,F*F**&F.\"\"$F,!\"\"F1/F.;,$F*F1F0" }
{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` =
 x+2*x^3/3-x^4/8;" "6#/%!G,(%\"xG\"\"\"*(\"\"#F'*$F&\"\"$F'F+!\"\"F'*&
F&\"\"%\"\")F,F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([3, ``],[`
`, ``],[-1, ``]);" "6#-%*PIECEWISEG6%7$\"\"$%!G7$F(F(7$,$\"\"\"!\"\"F(
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``=``(3+18-81/8)-(-1-2/3-1/8)" "6#/%!G,
&-F$6#,(\"\"$\"\"\"\"#=F**&\"#\")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 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=38/3" "6#/%!G*&\"#
Q\"\"\"\"\"$!\"\"" }{XPPEDIT 18 0 " ``= 12" "6#/%!G\"#7" }{TEXT -1 1 "
 " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "Int(1+2*x^2-x^3/2,x=-1..3);\n``=value(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$,(\"\"\"F'*&\"\"#F')%\"xGF)F'F'*&#F'F)F'*$)F+
\"\"$F'F'!\"\"/F+;F1F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"#Q\"
\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT 302 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 49 
"Find the area of the region bounded by the curve " }{XPPEDIT 18 0 "y \+
= 5*exp(-x/2);" "6#/%\"yG*&\"\"&\"\"\"-%$expG6#,$*&%\"xGF'\"\"#!\"\"F/
F'" }{TEXT -1 6 ", the " }{TEXT 301 1 "x" }{TEXT -1 29 " axis and the \+
vertical lines " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 5 "
 and " }{XPPEDIT 18 0 "x = 4;" "6#/%\"xG\"\"%" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT 303 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "f := x
 -> 5*exp(-x/2):\np1 := plot(f(x),x=0..4.5,y=-.5..6,color=red,thicknes
s=2):\na := 0: b := 4:\np2 := plot(f(x),x=a..b,color=grey,filled=true)
:\np3 := plot([[b,0],[b,f(b)]],color=black):\nplots[display]([p1,p2,p3
]);" }}{PARA 13 "" 1 "" {GLPLOT2D 403 334 334 {PLOTDATA 2 "6'-%'CURVES
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3#F2$\"3k31M:)*yh<F27$FbflF)7&FfflFafl7$$\"3y****\\nZ)H;#F2$\"3KH*RR[X
ap\"F27$FiflF)7&F]glFhfl7$$\"3YmmmJy*eC#F2$\"3phMH,_fE;F27$F`glF)7&Fdg
lF_gl7$$\"3')******R^bJBF2$\"3?`B+t5Se:F27$FgglF)7&F[hlFfgl7$$\"3f****
*\\5a`T#F2$\"3CYY81RX%\\\"F27$F^hlF)7&FbhlF]hl7$$\"3o****\\7RV'\\#F2$
\"3T(Gkd^!3N9F27$FehlF)7&FihlFdhl7$$\"3k*****\\@fke#F2$\"3W%GOF+;>P\"F
27$F\\ilF)7&F`ilF[il7$$\"3/LLL`4NnEF2$\"31PapF[`<8F27$FcilF)7&FgilFbil
7$$\"3#*******\\,s`FF2$\"3!yc)fh'[=E\"F27$FjilF)7&F^jlFiil7$$\"3[mm;zM
)>$GF2$\"3%[7N'pSU87F27$FajlF)7&FejlF`jl7$$\"3$*******pfa<HF2$\"3%y4(3
Nqgi6F27$FhjlF)7&F\\[mFgjl7$$\"3#HLLeg`!)*HF2$\"3>CBMxqt;6F27$F_[mF)7&
Fc[mF^[m7$$\"3w****\\#G2A3$F2$\"3oZVcgJsq5F27$Ff[mF)7&Fj[mFe[m7$$\"3;L
LL$)G[kJF2$\"3-CSLq'pv-\"F27$F]\\mF)7&Fa\\mF\\\\m7$$\"3#)****\\7yh]KF2
$\"3(>$3bnGaU)*F67$Fd\\mF)7&Fh\\mFc\\m7$$\"3xmmm')fdLLF2$\"3v6p,HWjU%*
F67$F[]mF)7&F_]mFj\\m7$$\"3bmmm,FT=MF2$\"3e%Gy*y'p/0*F67$Fb]mF)7&Ff]mF
a]m7$$\"3FLL$e#pa-NF2$\"3[;K:\"[Rwn)F67$Fi]mF)7&F]^mFh]m7$$\"3!*******
Rv&)zNF2$\"3'e]uJ8.'[$)F67$F`^mF)7&Fd^mF_^m7$$\"3ILLLGUYoOF2$\"3A]iC*o
0o)zF67$Fg^mF)7&F[_mFf^m7$$\"3_mmm1^rZPF2$\"3-!o*fTL^wwF67$F^_mF)7&Fb_
mF]_m7$$\"34++]sI@KQF2$\"3Iej\"R=W*etF67$Fe_mF)7&Fi_mFd_m7$$\"34++]2%)
38RF2$\"3!yq=)\\?InqF67$F\\`mF)7&F``mF[`m7$$\"\"%F)$\"37N1$=;knw'F67$F
c`mF)7\"-%&STYLEG6#%,PATCHNOGRIDG-Fjz6&F\\[l$\")=THvF_[lF_amF_am-F$6$7
$7$Fc`mF(Fb`m-Fjz6&F\\[lF)F)F)-%+AXESLABELSG6%Q\"x6\"Q\"yF[bm-%%FONTG6
#%(DEFAULTG-%%VIEWG6$;F($\"#X!\"\";$!\"&Fgbm$\"\"'F)" 1 2 0 1 10 0 2 
9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3
" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The required area is: " }
}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(5*exp(-x/2),x = \+
0 .. 4);" "6#-%$IntG6$*&\"\"&\"\"\"-%$expG6#,$*&%\"xGF(\"\"#!\"\"F0F(/
F.;\"\"!\"\"%" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = -10*exp(-x/2);" "6#/%!G,$*&\"#5\"\"\"-%$expG6#,$*&
%\"xGF(\"\"#!\"\"F0F(F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([4,
 ``],[0, ``]);" "6#-%*PIECEWISEG6$7$\"\"%%!G7$\"\"!F(" }{TEXT -1 1 " \+
" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(-10*exp(-
2))-(-10);" "6#/%!G,&-F$6#,$*&\"#5\"\"\"-%$expG6#,$\"\"#!\"\"F+F1F+,$F
*F1F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=10-10*exp(-2)" "6#/%!G,&\"#5\"
\"\"*&F&F'-%$expG6#,$\"\"#!\"\"F'F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 2 "  " }{TEXT 313 1 "~" }
{TEXT -1 9 " 8.6466. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 58 "Int(5*exp(-x/2),x=0..4);\n``=value(%);\n`
`=evalf[6](rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"
\"&\"\"\"-%$expG6#,$*&\"\"#!\"\"%\"xGF)F0F)F)/F1;\"\"!\"\"%" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"#5\"\"\"-%$expG6#!\"#F(!\"\"F'F(" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"'lY')!\"&" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 315 8 "Question" }
{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 49 "Find the area of the re
gion bounded by the curve " }{XPPEDIT 18 0 "y = sin*x;" "6#/%\"yG*&%$s
inG\"\"\"%\"xGF'" }{TEXT -1 6 ", the " }{TEXT 314 1 "x" }{TEXT -1 29 "
 axis and the vertical lines " }{XPPEDIT 18 0 "x = Pi/4;" "6#/%\"xG*&%
#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 2*Pi/3;
" "6#/%\"xG*(\"\"#\"\"\"%#PiGF'\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 
"" 0 "" {TEXT 316 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "f := x -> s
in(x):\np1 := plot(f(x),x=0..Pi,y=0..1,color=red,thickness=2):\na := P
i/4: b := 2*Pi/3:\np2 := plot(f(x),x=a..b,color=grey,filled=true):\np3
 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b)]]],color=black):\nplots[disp
lay]([p1,p2,p3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 504 295 295 {PLOTDATA 
2 "6(-%'CURVESG6%7S7$$\"\"!F)F(7$$\"3%)eD2LzxZo!#>$\"3v9-&QTFC%oF-7$$
\"3)\\$px*G*f!G\"!#=$\"3q>Km$*>5x7F37$$\"3+5@exGm]>F3$\"3/`P()fcJQ>F37
$$\"3[99!=3o^i#F3$\"3/)Q8J`>^f#F37$$\"35!\\D0[nkH$F3$\"3'yFM)o\")3PKF3
7$$\"37\"=Za&z%)=RF3$\"3b2`lK+J>QF37$$\"3edXa()oGjXF3$\"33g-l[Vb1WF37$
$\"3W%3**Hbm(H_F3$\"3OWs$HJ6Y*\\F37$$\"37PRr4)3T*eF3$\"3g4tHMSrebF37$$
\"3y\"[)yykYxlF3$\"32tCIi;N8hF37$$\"3[s'ocGo$zrF3$\"3iG>vb;KylF37$$\"3
UQ0;gr'p&yF3$\"3S'e5DeyJ2(F37$$\"3vFMt?$[t`)F3$\"3Cf#G%e3SPvF37$$\"3u
\"p30k@I>*F3$\"3C$=wEj'y^zF37$$\"3#*R7\\HeV)y*F3$\"3=)oD&\\m_)H)F37$$
\"3#G[))*)3W'\\5!#<$\"3Y4SV'zgCn)F37$$\"30'[@XS@'46Fdp$\"3oX3_YFIb*)F3
7$$\"3G9w$3G*Qz6Fdp$\"3=c,/>?tV#*F37$$\"3>%3*3tc9T7Fdp$\"3#*zOG!*[bh%*
F37$$\"3Ey0DBA!*38Fdp$\"3'*))o<xI,f'*F37$$\"3qjE.#[AMP\"Fdp$\"3W%*yQT$
\\e!)*F37$$\"3*R:_MgU2W\"Fdp$\"3!f%R/z#\\b\"**F37$$\"3\"Qu#)**[jD]\"Fd
p$\"3!>/b+VIn(**F37$$\"3=hw\"ymX#p:Fdp$\"274'Gxz)*****Fdp7$$\"3mwM!*4(
4&Q;Fdp$\"39vB0ZK3x**F37$$\"3CDL:iU!))p\"Fdp$\"3[oG=d;==**F37$$\"3o(R1
/.CRw\"Fdp$\"3Vz\\$>Q(39)*F37$$\"32Id)H7*>J=Fdp$\"3?VTu'[jGm*F37$$\"3G
fb7wY,(*=Fdp$\"3Sm0JN*4EZ*F37$$\"3cM5'zg%pg>Fdp$\"3ADf%RFx%\\#*F37$$\"
3**oJ8:.SJ?Fdp$\"3?af@Y>%y&*)F37$$\"3)**p!zPD$\\4#Fdp$\"3?%>'G5ccd')F3
7$$\"3k<!fhumF;#Fdp$\"3-mo\">-G%)H)F37$$\"3wak3@YBCAFdp$\"3cJr!4)G)*Rz
F37$$\"39/b<W_V\"H#Fdp$\"3l!*y#R$G%Q^(F37$$\"3#*z_V$zlYN#Fdp$\"3'4._Q1
p=3(F37$$\"3cmjYO*f2U#Fdp$\"3BSu)y%[5+mF37$$\"3)eq)=U!z`[#Fdp$\"3_nY*
\\Y775'F37$$\"3#Q'HHd#HIb#Fdp$\"3O<W\\nrm^bF37$$\"3z^r&[X%==EFdp$\"3o1
39l%\\$)*\\F37$$\"3'=BS\\0:[o#Fdp$\"33\"RnBm$e5WF37$$\"3[gN,?R*3v#Fdp$
\"3S5BwseM3QF37$$\"3@/0LMNh6GFdp$\"3O2NfUFBSKF37$$\"3w#oUX107)GFdp$\"3
!*=aVZ4buDF37$$\"3W46xe&[M%HFdp$\"3aOu_!z+&o>F37$$\"3H.6)e58)4IFdp$\"3
M.OA\"o%)RJ\"F37$$\"3Kt2RXCLtIFdp$\"3ls_wI6s?oF-7$$\"3!)***\\/l#fTJFdp
$\"3pawpOMzRJ!#E-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%*THICKNESSG6#\"
\"#-%)POLYGONSG6U7&7$$\"3o***z'\\;)R&yF3F)7$Fh[l$\"3E_4(H#y1rqF37$$\"3
&ppb#HdIR\")F3$\"3CEUpjp\"*psF37$F^\\lF)7&Fb\\lF]\\l7$$\"3e[\\F&okvQ)F
3$\"3:X52M`7QuF37$Fe\\lF)7&Fi\\lFd\\l7$$\"3GmA\"f%yvm')F3$\"3JWdfb<\"=
i(F37$F\\]lF)7&F`]lF[]l7$$\"3u(=\\+o,y%*)F3$\"3Wk&)pKbr+yF37$Fc]lF)7&F
g]lFb]l7$$\"31m4!>w4vA*F3$\"3)fexD/_E(zF37$Fj]lF)7&F^^lFi]l7$$\"3g1V!*
e\\$o[*F3$\"3'yYR$>,\\E\")F37$Fa^lF)7&Fe^lF`^l7$$\"31N'*pz6Nb(*F3$\"3Y
4#evg?+G)F37$Fh^lF)7&F\\_lFg^l7$$\"3#)ei$R50L+\"Fdp$\"3PvVX39_K%)F37$F
__lF)7&Fc_lF^_l7$$\"3Dq`?Jg)4.\"Fdp$\"3Y/$\\z*>1y&)F37$Ff_lF)7&Fj_lFe_
l7$$\"3I6,Kn#f%f5Fdp$\"3_f67d+\"4s)F37$F]`lF)7&Fa`lF\\`l7$$\"3m!G)=4&Q
X3\"Fdp$\"3f=*fNWy3%))F37$Fd`lF)7&Fh`lFc`l7$$\"3Te&HbzrF6\"Fdp$\"3UWz[
TvFp*)F37$F[alF)7&F_alFj`l7$$\"3k^rX557T6Fdp$\"3y(y***G0,\"4*F37$FbalF
)7&FfalFaal7$$\"3Ut;-K2Wo6Fdp$\"3>msOw7T,#*F37$FialF)7&F]blFhal7$$\"3+
f$ykl\\K>\"Fdp$\"3yc?F,$=dH*F37$F`blF)7&FdblF_bl7$$\"3!e!zAn)\\FA\"Fdp
$\"3yJfzYyS+%*F37$FgblF)7&F[clFfbl7$$\"3g'y\"G:/uZ7Fdp$\"3sV>X(*pp#[*F
37$F^clF)7&FbclF]cl7$$\"3=:\"[OO5oF\"Fdp$\"3q9Vn7Z'4d*F37$FeclF)7&Ficl
Fdcl7$$\"3;1bm$>UDI\"Fdp$\"3%pvo*[5NU'*F37$F\\dlF)7&F`dlF[dl7$$\"3+]-Z
*3u2L\"Fdp$\"37TRT()oK8(*F37$FcdlF)7&FgdlFbdl7$$\"33`\"Q0`dwN\"Fdp$\"3
YfP[[yrt(*F37$FjdlF)7&F^elFidl7$$\"3WYV.\"e2dQ\"Fdp$\"3RZ_%GA*>H)*F37$
FaelF)7&FeelF`el7$$\"3\"f%zm$Gm9T\"Fdp$\"39]`S\")zLt)*F37$FhelF)7&F\\f
lFgel7$$\"3QO\"QV_]#R9Fdp$\"33$>vM?.O\"**F37$F_flF)7&FcflF^fl7$$\"33\"
)exT06o9Fdp$\"35%4d!eWKZ**F37$FfflF)7&FjflFefl7$$\"3-lYzrKB$\\\"Fdp$\"
3;8frR\\$*p**F37$F]glF)7&FaglF\\gl7$$\"3<JYZ3lO?:Fdp$\"31;fwhoG()**F37
$FdglF)7&FhglFcgl7$$\"3ebT&pz(R[:Fdp$\"39no<M;\\(***F37$F[hlF)7&F_hlFj
gl7$$\"3'*=E+W4#ed\"Fdp$\"3Wi^PmP()****F37$FbhlF)7&FfhlFahl7$$\"30dR%)
[UN-;Fdp$\"39S]-,4-&***F37$FihlF)7&F]ilFhhl7$$\"3wFc1g\\\"=j\"Fdp$\"3G
yYS2%*Q\")**F37$F`ilF)7&FdilF_il7$$\"3kkjT>nGe;Fdp$\"3%**>wKc^<'**F37$
FgilF)7&F[jlFfil7$$\"3Cvr!G(4b'o\"Fdp$\"31?8F$3zI$**F37$F^jlF)7&FbjlF]
jl7$$\"3=%Htsei@r\"Fdp$\"3A'=u#)>W-!**F37$FejlF)7&FijlFdjl7$$\"3?R')zY
G;S<Fdp$\"3YW>!yU<p&)*F37$F\\[mF)7&F`[mF[[m7$$\"3?Q2!*3*3lw\"Fdp$\"33A
'3#fM44)*F37$Fc[mF)7&Fg[mFb[m7$$\"3qk,2N\"[Sz\"Fdp$\"3IkJ'**\\E=v*F37$
Fj[mF)7&F^\\mFi[m7$$\"3&Q.jBws4#=Fdp$\"3yk(=xb(o)o*F37$Fa\\mF)7&Fe\\mF
`\\m7$$\"3yzS*=Ng\"\\=Fdp$\"38'zI#HE1:'*F37$Fh\\mF)7&F\\]mFg\\m7$$\"3Y
gpy]$3j(=Fdp$\"3MIuH-2$p`*F37$F_]mF)7&Fc]mF^]m7$$\"3p\"HF25rS!>Fdp$\"3
kl$*>]5w\\%*F37$Ff]mF)7&Fj]mFe]m7$$\"3=253hRgJ>Fdp$\"3YZzH(4?hN*F37$F]
^mF)7&Fa^mF\\^m7$$\"3%G\"QH+Q!p&>Fdp$\"3!RO!4E%>QE*F37$Fd^mF)7&Fh^mFc^
m7$$\"3cyq6r-!f)>Fdp$\"3!=JqC1W2:*F37$F[_mF)7&F__mFj^m7$$\"3pX3Hg]$=,#
Fdp$\"3V1C6wt3V!*F37$Fb_mF)7&Ff_mFa_m7$$\"3)45E[&p[R?Fdp$\"3duGc._g@*)
F37$Fi_mF)7&F]`mFh_m7$$\"3S!e)>YL&f1#Fdp$\"3m>/-)4T*)z)F37$F``mF)7&Fd`
mF_`m7$$\"39++@)4&R%4#Fdp$\"3!QOg(eSDg')F37$Fg`mF)7\"-%&STYLEG6#%,PATC
HNOGRIDG-Fiz6&F[[l$\")=THvF^[lFcamFcam-F$6$7$7$$\"3!G[uRj\")R&yF3F(7$F
iam$\"3sva'=\"y1rqF3-Fiz6&F[[lF)F)F)-F$6$7$7$$\"3E&>$R-^R%4#FdpF(7$Fdb
m$\"3'fQWy.a-m)F3F^bm-%+AXESLABELSG6%Q\"x6\"Q\"yF]cm-%%FONTG6#%(DEFAUL
TG-%%VIEWG6$;F($\"+aEfTJ!\"*;F($\"\"\"F)" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The required area is: " }
}{PARA 257 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "Int(sin*x,x = Pi/4 \+
.. 2*Pi/3);" "6#-%$IntG6$*&%$sinG\"\"\"%\"xGF(/F);*&%#PiGF(\"\"%!\"\"*
(\"\"#F(F-F(\"\"$F/" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "`` = -cos*x;" "6#/%!G,$*&%$cosG\"\"\"%\"xGF(!\"\"" }
{TEXT -1 2 "  " }{XPPEDIT 18 0 "PIECEWISE([2*Pi/3, ``],[Pi/4, ``]);" "
6#-%*PIECEWISEG6$7$*(\"\"#\"\"\"%#PiGF)\"\"$!\"\"%!G7$*&F*F)\"\"%F,F-
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(-cos(2*Pi/3))-(-cos(Pi/4));" "6
#/%!G,&-F$6#,$-%$cosG6#*(\"\"#\"\"\"%#PiGF.\"\"$!\"\"F1F.,$-F*6#*&F/F.
\"\"%F1F1F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=cos(Pi/4)-cos(2*Pi/3)" "
6#/%!G,&-%$cosG6#*&%#PiG\"\"\"\"\"%!\"\"F+-F'6#*(\"\"#F+F*F+\"\"$F-F-
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/sqrt(2)-(-1/2)" "6#/%!G,&*&\"\"\"F
'-%%sqrtG6#\"\"#!\"\"F',$*&F'F'F+F,F,F," }{TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
``=1/2+1/sqrt(2)" "6#/%!G,&*&\"\"\"F'\"\"#!\"\"F'*&F'F'-%%sqrtG6#F(F)F
'" }{TEXT -1 2 " \n" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 317 1 "
~" }{TEXT -1 8 " 1.2071 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Int(sin(x),x=Pi/4..2*Pi/3);\n``=val
ue(%);\n``=evalf[6](rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Int
G6$-%$sinG6#%\"xG/F);,$*&\"\"%!\"\"%#PiG\"\"\"F1,$*(\"\"#F1\"\"$F/F0F1
F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"#!\"\"F'#\"\"\"F'F*F
)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"'527!\"&" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 319 8 "Question" }
{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 49 "Find the area of the re
gion bounded by the curve " }{XPPEDIT 18 0 "y = 3/(2*x+1);" "6#/%\"yG*
&\"\"$\"\"\",&*&\"\"#F'%\"xGF'F'F'F'!\"\"" }{TEXT -1 6 ", the " }
{TEXT 318 1 "x" }{TEXT -1 29 " axis and the vertical lines " }
{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT 320 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 229 "f := x -> 3/(2*x+1):\np
1 := plot(f(x),x=-.2..3,y=0..4,color=red,thickness=2):\na := 0: b := 2
:\np2 := plot(f(x),x=a..b,color=grey,filled=true):\np3 := plot([[[a,0]
,[a,f(a)]],[[b,0],[b,f(b)]]],color=black):\nplots[display]([p1,p2,p3])
;" }}{PARA 13 "" 1 "" {GLPLOT2D 403 334 334 {PLOTDATA 2 "6(-%'CURVESG6
%7Y7$$!35+++++++?!#=$\"\"&\"\"!7$$!3!pmmmwAc#=F*$\"3`NJF4jLDZ!#<7$$!3S
LLLLbC^;F*$\"3oP02StFzWF37$$!3!********HooZ\"F*$\"3_z7YCdddUF37$$!3qmm
mm5\\-8F*$\"3#yIi/L&ycSF37$$!3?++++nT!***!#>$\"3Q'fA3y,\"\\PF37$$!3WLL
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$6$7$7$FccmF`]l7$Fccm$\"3w**************fF*Fddm-%+AXESLABELSG6%Q\"x6\"
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1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 
";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Th
e required area is: " }}{PARA 257 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 
18 0 "Int(3/(2*x+1),x = 0 .. 2);" "6#-%$IntG6$*&\"\"$\"\"\",&*&\"\"#F(
%\"xGF(F(F(F(!\"\"/F,;\"\"!F+" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 3/2;" "6#/%!G*&\"\"$\"\"\"\"\"#!\"
\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(2*x+1);" "6#-%#lnG6#,&*&\"\"#\"
\"\"%\"xGF)F)F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([2, ``],[
0, ``]);" "6#-%*PIECEWISEG6$7$\"\"#%!G7$\"\"!F(" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = ``(3*ln*5/2)-3*ln*1/2;" "6#/%!G,&-F$6#**\"\"$\"\"
\"%#lnGF*\"\"&F*\"\"#!\"\"F***F)F*F+F*F*F*F-F.F." }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 3*ln*5/2;" "6#/%!G**\"\"$\"\"\"%#lnGF'\"\"&F'\"\"#
!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 
0 "" {TEXT -1 1 " " }{TEXT 321 1 "~" }{TEXT -1 8 " 2.4141 " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Int(
3/(2*x+1),x=0..2);\n``=value(%);\n``=evalf[6](rhs(%));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"$\"\"\",&*&\"\"#F)%\"xGF)F)F)F)!
\"\"F)/F-;\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\"\"$\"
\"#\"\"\"-%#lnG6#\"\"&F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"'
;9C!\"&" }}}{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 "T
asks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "
" {TEXT -1 36 "In each of the following questions: " }}{PARA 0 "" 0 "
" {TEXT -1 54 "(a) Draw a picture to illustrate the region involved, \+
" }}{PARA 0 "" 0 "" {TEXT -1 77 "(b) Find the area of the region by us
ing the Fundamental Theorem of Calculus " }}{PARA 0 "" 0 "" {TEXT -1 
119 "(c) Calculate some approximations for the area using left and/or \+
right hand approximating sums of areas of rectangles. " }}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 49 "Find t
he area of the region bounded by the curve " }{XPPEDIT 18 0 "y = x^2-3
*x+4;" "6#/%\"yG,(*$%\"xG\"\"#\"\"\"*&\"\"$F)F'F)!\"\"\"\"%F)" }{TEXT 
-1 6 ", the " }{TEXT 322 1 "x" }{TEXT -1 29 " axis and the vertical li
nes " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 2 ". " }}{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 49 "Find the area of the region bounded by the curve " }
{XPPEDIT 18 0 "y = 2-2*x+x^4/4;" "6#/%\"yG,(\"\"#\"\"\"*&F&F'%\"xGF'!
\"\"*&F)\"\"%F,F*F'" }{TEXT -1 6 ", the " }{TEXT 323 1 "x" }{TEXT -1 
29 " axis and the vertical lines " }{XPPEDIT 18 0 "x = -1;" "6#/%\"xG,
$\"\"\"!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"
\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "_________________
____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
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" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 37 "_____________________________________" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 49 "Find the area of the
 region bounded by the curve " }{XPPEDIT 18 0 "y = 3/5;" "6#/%\"yG*&\"
\"$\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(2*x)" "6#-%$e
xpG6#*&\"\"#\"\"\"%\"xGF(" }{TEXT -1 6 ", the " }{TEXT 324 1 "x" }
{TEXT -1 29 " axis and the vertical lines " }{XPPEDIT 18 0 "x = -1;" "
6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 1;" "6#
/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{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 49 "Find the area of \+
the region bounded by the curve " }{XPPEDIT 18 0 "y = cos*x;" "6#/%\"y
G*&%$cosG\"\"\"%\"xGF'" }{TEXT -1 6 ", the " }{TEXT 325 1 "x" }{TEXT 
-1 29 " axis and the vertical lines " }{XPPEDIT 18 0 "x = -Pi/4;" "6#/
%\"xG,$*&%#PiG\"\"\"\"\"%!\"\"F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "
x = Pi/3;" "6#/%\"xG*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________
_________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "
;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" 
{TEXT -1 49 "Find the area of the region bounded by the curve " }
{XPPEDIT 18 0 "y = 4/(3*x+2);" "6#/%\"yG*&\"\"%\"\"\",&*&\"\"$F'%\"xGF
'F'\"\"#F'!\"\"" }{TEXT -1 6 ", the " }{TEXT 326 1 "x" }{TEXT -1 29 " \+
axis and the vertical lines " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" 
}{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 3;" "6#/%\"xG\"\"$" }{TEXT -1 
2 ". " }}{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 25 "Code for drawing pic
tures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 
0 "" 0 "" {TEXT -1 48 "Code for Fundamental Theorem of Calculus pictur
e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 754 "fn := x -> x^2/2+2: x := 'x':\np1 := plot(fn(x),x=-0
.5..4.3,y=-0.5..10):\ncrv := op(op(1,op(1,plot(fn(x),x=0..3)))):\np2 :
= plot([[[3,0],[3,fn(3.5)]],[[3.5,0],[3.5,fn(3.5)]],\n  [[3,fn(3)],[3.
5,fn(3)]],[[3,fn(3.5)],[3.5,fn(3.5)]]],linestyle=2,\n    color=COLOR(R
GB,0.1,0.1,0.1)):\np3 := plots[polygonplot]([[3,0],[3,fn(3)],[3.5,fn(3
)],[3.5,0]],\n                 style=PATCHNOGRID,color=COLOR(RGB,.6,.6
,.6)):\np4 := plots[polygonplot]([[1.5,0],[0,0],crv,[3,0]],\n         \+
        style=PATCHNOGRID,color=COLOR(RGB,.9,.9,.9)):\nt1:=plots[textp
lot]([4.3,9.4,`y = f(x)`],color=red):\nt2:=plots[textplot]([[3,-0.3,`x
`],[3.5,-0.3,`x+h`],\n[-0.1,9.4,`y`],[1.8,2,`A(x)`],\n[2.5,6.5,`(x,f(x
))`],[4.1,8,`(x+h,f(x+h))`] ]):\nplots[display]([p1,p2,p3,p4,t1,t2],ti
ckmarks=[0,0]); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 42 "Co
de for general definite integral picture" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 512 "fn := x -> x^2/2+2: \+
x := 'x':\np1 := plot(fn(x),x=-.5..4.3,y=-.5..10):\ncrv := op(op(1,op(
1,plot(fn(x),x=1..3)))):\np2 := plot([[[1,0],[1,fn(1)]],[[3,0],[3,fn(3
)]]],linestyle=1,\n    color=COLOR(RGB,.1,.1,.1)):\np3 := plots[polygo
nplot]([[2,0],[1,0],crv,[3,0]],\n                 style=PATCHNOGRID,co
lor=COLOR(RGB,.7,.7,.7)):\nt1:=plots[textplot]([4.3,9.4,`y = f(x)`],co
lor=red):\nt2:=plots[textplot]([[4.2,-.3,`x`],[-.2,9.4,`y`],[1,-.25,`x
 = a`],\n  [3,-.25,`x = b`]]):\nplots[display]([p1,p2,p3,t1,t2],tickma
rks=[0,0]); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 26 "Code \+
for left sum picture " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 445 "f \+
:= x -> x^2/2+2:\np1 := plot(f(x),x=-0.1..2.1,y=-0.2..4.4,color=red):
\np2 :=student[leftbox](f(x),x=0.5..2,15,color=red,shading=gray,\n    \+
    tickmarks=[0,0]):\np3 := plot([[2,0],[2,f(2)]],linestyle=2,color=C
OLOR(RGB,.1,.1,.1)):\nt1 := plots[textplot]([1.9,4.3,`y = f(x)`],color
=red):\nt2 := plots[textplot]([[0.5,-0.1,`a=x`],[0.59,-0.18,`0`],\n   \+
[2.,-0.1,`b=x`],[2.08,-0.18,`n`],[-0.05,4.3,`y`]]):\nplots[display]([p
1,p2,p3,t1,t2],tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "
" {TEXT -1 27 "Code for right sum picture " }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 446 "f := x -> x^2/2+2:\np1 := plot(f(x),x=-0.1..2.1,y=
-0.2..4.4,color=red):\np2 :=student[rightbox](f(x),x=0.5..2,15,color=r
ed,shading=gray,\n        tickmarks=[0,0]):\np3 := plot([[2,0],[2,f(2)
]],linestyle=2,color=COLOR(RGB,.1,.1,.1)):\nt1 := plots[textplot]([1.9
,4.3,`y = f(x)`],color=red):\nt2 := plots[textplot]([[0.5,-0.1,`a=x`],
[0.59,-0.18,`0`],\n   [2.,-0.1,`b=x`],[2.08,-0.18,`n`],[-0.05,4.3,`y`]
]):\nplots[display]([p1,p2,p3,t1,t2],tickmarks=[0,0]);" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 0 "" 0 "" {TEXT -1 21 "Code for y=x example " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 421 "p1 \+
:= plot(x,x=-0.1..1.1,y=-0.1..1.1,color=red):\np2 :=student[leftbox](x
,x=0..1,15,color=red,shading=gray,\n        tickmarks=[0,0]):\np3 := p
lot([[1,0],[1,1]],linestyle=2,color=COLOR(RGB,.1,.1,.1)):\nt1 := plots
[textplot]([.9,1.05,`y = x`],color=red):\nt2 := plots[textplot]([[0.06
,-0.05,`x  = 0`],[0.043,-0.08,`0`],\n   [1,-0.05,`x  = 1`],[.98,-0.075
,`n`],[-0.05,4.3,`y`]]):\nplots[display]([p1,p2,p3,t1,t2],tickmarks=[0
,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 29 "Code for m
iddle sum picture  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 468 "f := x -> 1/2+sin(x/2)-cos(5/2*x)/4:\np1 :
= plot(f(x),x=-0.1..2.3,y=-0.1..1.5,color=red):\np2 :=student[middlebo
x](f(x),x=0.5..2,15,color=red,shading=gray,\n        tickmarks=[0,0]):
\np3 := plot([[2,0],[2,f(2)]],linestyle=2,color=COLOR(RGB,.1,.1,.1)):
\nt1 := plots[textplot]([2.2,1.4,`y = f(x)`],color=red):\nt2 := plots[
textplot]([[0.5,-0.05,`a=x`],[0.6,-0.09,`0`],\n   [2.,-0.05,`b=x`],[2.
085,-0.09,'n'],[-0.05,1.45,`y`]]):\nplots[display]([p1,p2,p3,t1,t2],ti
ckmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 23 "Cod
e for area element  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1019 "f := x -> 2+sin(x/2)-cos(5/2*x)/4:\na :
= .7: b := 3.4:\nc := 2: d := 2.1: e := (c+d)/2:\np1 := plot(f(x),x=.5
..3.6,y=0..4,thickness=2):\np2 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b
)]]],\n                color=COLOR(RGB,.4,.4,.4),linestyle=1):\np3 := \+
plot(f(x),x=c..d,adaptive=false,numpoints=5,\n               filled=tr
ue,color=COLOR(RGB,.6,.6,.6)):\np4 := plot(f(x),x=a..b,adaptive=false,
numpoints=25,\n               filled=true,color=COLOR(RGB,.8,.8,.8)):
\np5 := plot([[[c,0],[c,f(c)]],[[d,0],[d,f(d)]]],color=black):\np6 := \+
plot([[0,f(e)],[e,f(e)]],color=black,linestyle=3):\np7 := plottools[ar
row]([c-.2,1],[c,1],0,.1,.3,arrow,color=black):\np8 := plottools[arrow
]([d+.2,1],[d,1],0,.1,.3,arrow,color=black):\nt1 := plots[textplot]([3
.2,3.4,`y = f(x)`],color=red):\nt2 := plots[textplot]([[a,-.14,`x=a`],
[b,-.14,`x=b`],\n         [2.46,1,`x`],[e,-.14,`x`],[-.15,f(e),'f(x)']
],color=black):\nt3 := plots[textplot]([2.4,1,'d'],font=[SYMBOL,10]):
\nplots[display]([p1,p2,p3,p4,p5,p6,p7,p8,t1,t2,t3],tickmarks=[0,0],vi
ew=[-.2..4,-.2..4]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 
25 "Code for sliced region.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 572 "f := x -> 2+sin(x/2)-cos(5/
2*x)/4:\na := .7: b := 3.4:\nn := 40: h := (b-a)/n:\np1 := plot(f(x),x
=.5..3.6,y=0..4,thickness=2):\np2 := plot([seq([[a+i*h,0],[a+i*h,f(a+i
*h)]],i=0..n)],\n        color=COLOR(RGB,.4,.4,.4)):\np3 := plot(f(x),
x=a..b,adaptive=false,numpoints=25,\n               filled=true,color=
COLOR(RGB,.8,.8,.8)):\nn := 'n': h := 'h':\nt1 := plots[textplot]([3.2
,3.4,`y = f(x)`],color=red):\nt2 := plots[textplot]([[a,-.15,`x=a`],[b
,-.15,`x=b`],[-0.08,3.9,`y`]],color=black):\na := 'a': b := 'b': \nplo
ts[display]([p1,p2,p3,t1,t2],tickmarks=[0,0],view=[-.2..4,-.2..4]);" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 52 "Code for sliced regio
n using genuine rectangles. .  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 444 "f := x -> 2+sin(x/2)-cos(5/
2*x)/4:\na := .7: b := 3.4:\np1 := plot(f(x),x=.5..3.6,y=0..4,thicknes
s=2):\np2 :=student[middlebox](f(x),x=a..b,40,color=red,shading=COLOR(
RGB,.8,.8,.8),\n        tickmarks=[0,0],thickness=2):\nt1 := plots[tex
tplot]([3.2,3.4,`y = f(x)`],color=red):\nt2 := plots[textplot]([[a,-.1
5,`x=a`],[b,-.15,`x=b`],[-0.08,3.9,`y`]],color=black):\na := 'a': b :=
 'b': \nplots[display]([p1,p2,t1,t2],tickmarks=[0,0],view=[-.2..4,-.2.
.4]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 
1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }