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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f := x -> x;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "n := 10;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 26 "g := x -> trunc(n*f(x))/n;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 7 "b := 1;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "plot(\{g(x),f(x)\}, x=0..b);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 45 "volEst_1 := evalf(int('2*Pi*x*g(x)',x=0..1));" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "volEst_2 := evalf(sum(2*Pi*(i/n)*f
(i/n)*(1/n), i=0..(n-1)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "vol :
= evalf(int(2*Pi*x*f(x),x=0..1));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
48 "PerCentErr_1 := evalf(100*(vol - volEst_1)/vol);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 40 "PerCentErr_2 := 100*(vol -volEst_2)/vol;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "vo
lEst_1 is the volume of revolution of the step function g(x)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "volEst_2 \+
is the volume of the sum of the shells obtained using g(x)." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "vol is the actu
al volume obtained by the volume of revolution integral for f(x)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The respe
ctive per cent errors are also obtained." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "No
w we graph the surface of our volume of revolution but view it from be
low." }}{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 43 "plot3d(g(sqrt(1-x^2-y^2)),x=-1..1,y=-1..1);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 41 "plot3d(g(sqrt(x^2+y^2)),x=-1..1,y=-1..1);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "f := x -> x^2;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "n := 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "
g := x -> trunc(n*x^2)/n;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 
"b := 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(\{g(x),f(x
)\}, x=0..b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "volEst_1 :
= evalf(int('2*Pi*x*g(x)',x=0..1));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
60 "volEst_2 := evalf(sum(2*Pi*(i/n)*f(i/n)*(1/n), i=0..(n-1)));" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "vol := evalf(int(2*Pi*x*f(x),x=0..1
));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "PerCentErr_1 := evalf(100*(v
ol - volEst_1)/vol);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "PerCentErr_
2 := 100*(vol -volEst_2)/vol;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "
volEst_1 is the volume of revolution of the step function g(x)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "volEst_2 \+
is the volume of the sum of the shells obtained using g(x)." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "vol is the actu
al volume obtained by the volume of revolution integral for f(x)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The respe
ctive per cent errors are also obtained." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Be
low we plot a representation of the surface for the volume of revoluti
on from below:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 43 "plot3d(g(sqrt(1-x^2-y^2)),x=-1..1,y=-1..1);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "19" 0 }
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