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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 25 "Calculus Single Variable
 " }{TEXT 259 2 " \n" }{TEXT 257 35 "Brian E. Blank and Steven G. Kran
tz" }{TEXT 258 46 "\n\nSection 8.1\nVolumes of Solids of Revolution\n
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 29 "Examples from the Exercises
\n\n" }}{PARA 0 "" 0 "" {TEXT 276 30 "Example (Exercise 1, Page 573)" 
}{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The region bounded above
 by " }{XPPEDIT 18 0 "y = 3*x^(1/3);" "6#/%\"yG*&\"\"$\"\"\")%\"xG*&F'
F'F&!\"\"F'" }{TEXT -1 16 ",  below by the " }{XPPEDIT 18 0 "x;" "6#%
\"xG" }{TEXT -1 43 "-axis, and laterally by the vertical lines " }
{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 
18 0 "x = 8;" "6#/%\"xG\"\")" }{TEXT -1 23 "  is rotated about the " }
{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 28 "-axis. Calculate the volume
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "`Volume = `*Int(Pi*(3*x^(1/3
))^2, x = 1 .. 8) = simplify(int(Pi*(3*x^(1/3))^2, x = 1 .. 8));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%*Volume~=~G\"\"\"-%$IntG6$,$*(\"\"
*F&%#PiGF&)%\"xG#\"\"#\"\"$F&F&/F/;F&\"\")F&,$*(\"$P)F&\"\"&!\"\"F-F&F
&" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 30 "Ex
ample (Exercise 3, Page 573)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 28 "The region bounded above by " }{XPPEDIT 18 0 "y = sqrt(sin(x));
" "6#/%\"yG-%%sqrtG6#-%$sinG6#%\"xG" }{TEXT -1 18 " and below by the \+
" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 10 "-axis for " }{XPPEDIT 
18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= Pi;" "6#1%!G%#PiG
" }{TEXT -1 24 "   is rotated about the " }{XPPEDIT 18 0 "x;" "6#%\"xG
" }{TEXT -1 29 "-axis. \nCalculate the volume." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 86 "`Volume = `*Int(Pi*sqrt(sin(x))^2, x = 0 .. Pi) = \+
int(Pi*sqrt(sin(x))^2, x = 0 .. Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/*&%*Volume~=~G\"\"\"-%$IntG6$*&%#PiGF&-%$sinG6#%\"xGF&/F/;\"\"!F+F&
,$*&\"\"#F&F+F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 30 "Example (Exercise 5, Pag
e 573)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The region bound
ed between the " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 11 "-axis and
  " }{XPPEDIT 18 0 "x = y^3;" "6#/%\"xG*$)%\"yG\"\"$\"\"\"" }{TEXT -1 
7 "   for " }{XPPEDIT 18 0 "0 <= y;" "6#1\"\"!%\"yG" }{XPPEDIT 18 0 "`
` <= 1;" "6#1%!G\"\"\"" }{TEXT -1 24 "   is rotated about the " }
{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 29 "-axis. \nCalculate the volu
me." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "`Volume = `*Int(Pi*(y^3)^2,
 y = 0..1) = int(Pi*(y^3)^2, y = 0..1);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*&%*Volume~=~G\"\"\"-%$IntG6$*&%#PiGF&)%\"yG\"\"'F&/F-;\"\"!F&F
&,$*&\"\"(!\"\"F+F&F&" }}}{PARA 3 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "
" 0 "" {TEXT 279 30 "Example (Exercise 7, Page 573)" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 31 "The region bounded between the " }
{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 12 "-axis  and  " }{XPPEDIT 18 
0 "y = ln(x);" "6#/%\"yG-%#lnG6#%\"xG" }{TEXT -1 7 "   for " }
{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= exp(1);
" "6#1%!G-%$expG6#\"\"\"" }{TEXT -1 24 "   is rotated about the " }
{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 29 "-axis. \nCalculate the volu
me." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "`Volume = `*Int(Pi*(exp(y))
^2, y = 0..1) = factor(int(Pi*(exp(y))^2, y = 0..1));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/*&%*Volume~=~G\"\"\"-%$IntG6$*&%#PiGF&)-%$expG6#%
\"yG\"\"#F&/F0;\"\"!F&F&,$*&#F&F1F&*&F+F&,&-F.6#F1F&F&!\"\"F&F&F&" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 30 "Example \+
(Exercise 9, Page 573)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "
The region bounded between  " }{XPPEDIT 18 0 "y = x;" "6#/%\"yG%\"xG" 
}{TEXT -1 8 "   and  " }{XPPEDIT 18 0 "y = x^2;" "6#/%\"yG*$)%\"xG\"\"
#\"\"\"" }{TEXT -1 7 "   for " }{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG
" }{XPPEDIT 18 0 "`` <= 1;" "6#1%!G\"\"\"" }{TEXT -1 24 "   is rotated
 about the " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 29 "-axis. \nCalc
ulate the volume." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "`Volume = `*
Pi*Int((x)^2-(x^2)^2, x = 0..1) = Pi*int((x)^2-(x^2)^2, x = 0..1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*(%*Volume~=~G\"\"\"%#PiGF&-%$IntG6$,
&*$)%\"xG\"\"#F&F&*$)F.\"\"%F&!\"\"/F.;\"\"!F&F&,$*(F/F&\"#:F3F'F&F&" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 281 31 "Example (Exercise 11, Page 573)" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The region bounded above by  " 
}{XPPEDIT 18 0 "y = 4-x^2;" "6#/%\"yG,&\"\"%\"\"\"*$)%\"xG\"\"#F'!\"\"
" }{TEXT -1 17 "   and below by  " }{XPPEDIT 18 0 "y = x+2;" "6#/%\"yG
*$,&%\"xG\"\"\"\"\"#F(F(" }{TEXT -1 24 "   is rotated about the " }
{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 29 "-axis. \nCalculate the volu
me." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "solve(4-x^2 = x+2, x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$!\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 316 "pl1 := plot([4-x^2 , x+2],x = -2 .. 1, thickness=[
2,2], color=COLOR(RGB,.30,.70,.99),view=[-2..1,0..4]):\nfor j from 1 t
o 99 do\nfill[j] := plot([-2+3*j/100,y,y=3*j/100..4-(-2+3*j/100)^2], t
hickness=3, color=COLOR(RGB,.30,.70,.99),view=[-2..1,0..4]):\nend do:
\nplots[display](pl1, seq(fill[j],j=1..99), tickmarks=[5,5]);" }}
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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" "Cur
ve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 2
2" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "
Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curv
e 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40" "Curve 41
" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47" "C
urve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53" "Curve
 54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curve 59" "Curve 60
" "Curve 61" "Curve 62" "Curve 63" "Curve 64" "Curve 65" "Curve 66" "C
urve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Curve 72" "Curve
 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "Curve 78" "Curve 79
" "Curve 80" "Curve 81" "Curve 82" "Curve 83" "Curve 84" "Curve 85" "C
urve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90" "Curve 91" "Curve
 92" "Curve 93" "Curve 94" "Curve 95" "Curve 96" "Curve 97" "Curve 98
" "Curve 99" "Curve 100" "Curve 101" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 579 "outerSurface := plot3d([x,(4-x^2)*cos(t),(4-x^2)*sin(t)],x=-2
..1,t=0..2*Pi,style=WIREFRAME):\ninnerSurface := plot3d([x,(x+2)*cos(t
),(x+2)*sin(t)],x=-2..1,t=0..2*Pi,style=WIREFRAME):\nwasher1 := plot3d
([.5,r*cos(t),r*sin(t)],t=0..2*Pi,r=.5+2..4-(.5)^2, color=NAVY, style=
PATCHNOGRID):\nwasher2 := plot3d([.54,r*cos(t),r*sin(t)],t=0..2*Pi,r=.
54+2..4-(.54)^2, color=NAVY, style=PATCHNOGRID):\nwasher3 := plot3d([.
56,r*cos(t),r*sin(t)],t=0..2*Pi,r=.56+2..4-(.56)^2, color=NAVY, style=
PATCHNOGRID):\nplots[display](outerSurface,innerSurface,washer1,washer
2,washer3,orientation=[-80,86]); " }}{PARA 13 "" 1 "" {GLPLOT3D 375 
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BBD1AD24631A2F6FFF4F=-%+PROJECTIONG6%$!#!)\"\"!$\"#')FK\"\"\"" 1 2 0 
1 10 0 2 1 1 1 2 1.000000 86.000000 -80.000000 1 0 "Curve 1" "Curve 2
" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "`Volume = `*Pi*Int((4-x^2)^2
-(x+2)^2, x = -2..1) = Pi*int((4-x^2)^2-(x+2)^2, x = -2..1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*(%*Volume~=~G\"\"\"%#PiGF&-%$IntG6$,&*$),&
\"\"%F&*$)%\"xG\"\"#F&!\"\"F3F&F&*$),&F2F&F3F&F3F&F4/F2;!\"#F&F&,$*(\"
$3\"F&\"\"&F4F'F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 31 "Example (Exercise 21, Page 574
)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The region bounded ab
ove by " }{XPPEDIT 18 0 "y = 4;" "6#/%\"yG\"\"%" }{TEXT -1 21 ", on th
e left by the " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 30 "-axis, and
 below by the curve " }{XPPEDIT 18 0 "x = sqrt(y);" "6#/%\"xG-%%sqrtG6
#%\"yG" }{TEXT -1 24 "  is rotated about the \n" }{XPPEDIT 18 0 "x;" "
6#%\"xG" }{TEXT -1 28 "-axis. Calculate the volume." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 8 "Solution" }{TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 272 20 "Region to be rotated" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 293 "pl1 :=
 plot([x^2,4],x = 0 .. 2, thickness=[2,2], color=COLOR(RGB,.30,.70,.99
),view=[0..2,0..4]):\nfor j from 0 to 99 do\nfill[j] := plot([2*j/100,
y,y=(2*j/100)^2..4], thickness=3, color=COLOR(RGB,.30,.70,.99),view=[0
..2,0..4]):\nend do:\nplots[display](pl1, seq(fill[j],j=0..99), tickma
rks=[5,5]);" }}{PARA 13 "" 1 "" {GLPLOT2D 335 182 182 {PLOTDATA 2 "6eq
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ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1
8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "
Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curv
e 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37
" "Curve 38" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "C
urve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve
 50" "Curve 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56
" "Curve 57" "Curve 58" "Curve 59" "Curve 60" "Curve 61" "Curve 62" "C
urve 63" "Curve 64" "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve
 69" "Curve 70" "Curve 71" "Curve 72" "Curve 73" "Curve 74" "Curve 75
" "Curve 76" "Curve 77" "Curve 78" "Curve 79" "Curve 80" "Curve 81" "C
urve 82" "Curve 83" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve
 88" "Curve 89" "Curve 90" "Curve 91" "Curve 92" "Curve 93" "Curve 94
" "Curve 95" "Curve 96" "Curve 97" "Curve 98" "Curve 99" "Curve 100" "
Curve 101" "Curve 102" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 274 39 "Cutaway View of the Solid of revolution" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 419 "pl3
d1 := plot3d([4*cos(t),y,4*sin(t)],t = Pi/2 ..2*Pi, y = 0..2):\npl3d2 \+
:= plot3d([y^2*cos(t),y,y^2*sin(t)],t = Pi/2 ..2*Pi, y = 0..2):\npl3d3
 := plot3d([r*cos(t),0,r*sin(t)],t = Pi/2 ..2*Pi, r = 0..4, style=PATC
HNOGRID):\npl3d4 := plot3d([0,y,z], z = y^2..4,y = 0 ..2,style=PATCHNO
GRID):\npl3d5 := plot3d([x,y,0], x = y^2..4,y = 0 ..2,style=PATCHNOGRI
D):\nplots[display](pl3d1,pl3d2,pl3d3,pl3d4,pl3d5,orientation=[18,56])
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