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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 256 
"" 0 "" {TEXT -1 31 "Lesson 2:  Solids of Revolution" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning,
 the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 16 "with(plottools):" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 10 "Example 1 " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Find th
e volume of the solid obtained by rotating the region bounded by the g
iven curves about the " }}{PARA 0 "" 0 "" {TEXT -1 60 "given axis. Ske
tch the region and a typical disk or washer. " }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "y^2;" "6#*$%\"yG\"\"#" }{TEXT -1 31 " =
 x, x = 2y; about the y-axis " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f:= x -> x/2; " }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,$9$#\"
\"\"\"\"#F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "g1:= x -
> sqrt(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g1G%%sqrtG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g2:= x -> -sqrt(x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g2GR6#%\"xG6\"6$%)operatorG%&arrowG
F(,$-%%sqrtG6#9$!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Le
t's determine where the functions intersect. " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "solve( g1(x)=f(x), x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"\"%\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "We h
ave an intersection when x = 4 and y = 2. Let's plot. " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a:=plot(f(x), x = 0..4, color = red
, thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "b:= plot
(g1(x), x = 0..4, color = blue, thickness=2):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 53 "c:= plot(g2(x), x = 0..4, color = blue, thicknes
s=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "d:= seq( plot([0 \+
+ 4 * i/20, t, t = f( 0 + 4 * i/20)..g1(0 + 4 *i/20)], color = magenta
), i = 1..20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "display(
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{PARA 0 "" 0 "" {TEXT -1 35 " The area of a washer is given by: " }
{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 4 " (4 " }{XPPEDIT 18 0 "y^2;
" "6#*$%\"yG\"\"#" }{TEXT -1 5 " ) - " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" 
}{TEXT -1 3 " ( " }{XPPEDIT 18 0 "y^4;" "6#*$%\"yG\"\"%" }{TEXT -1 2 "
 )" }}{PARA 0 "" 0 "" {TEXT -1 45 " Hence the volume of the solid is p
recisely: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "int(Pi(4*
y^2-y^4),y = 0 .. 2);" "6#-%$intG6$-%#PiG6#,&*&\"\"%\"\"\"*$%\"yG\"\"#
F,F,*$F.F+!\"\"/F.;\"\"!F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 34 "We have maple do the integration. " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int(Pi * (4 * y^2 - \+
y^4), y = 0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"#k\"#:" 
}}}{PARA 0 "" 0 "" {TEXT -1 33 " Thus the volume of the solid is " }
{XPPEDIT 18 0 "64/15;" "6#*&\"#k\"\"\"\"#:!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 14 " cubic units. " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 9 "Example 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 " Find the volum
e of the solid obtained by rotating the region bounded by the given " 
}}{PARA 0 "" 0 "" {TEXT -1 25 " curve about the x-axis. " }}{PARA 0 "
" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 36 " y = sec(x), y = \+
1/2, x = -1, x = 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 48 "Sketch the region. Draw a typical cross setion. " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:= x -> sec(x);" }}{PARA 
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "b:= plot(1/2, x = -1..1, color = br
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eq( plot([-1 + 2*i /30, t, t = 1/2..sec(-1 + 2 * i/30)], color = red),
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LTG-%%VIEWG6$;FicqF_am;F_u$\"0$4o<d\"3&=!#9" 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" "Curve 21" "Curve 22" "Curve 23" "C
urve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve
 30" "Curve 31" "Curve 32" "Curve 33" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 93 " We take the region shaded in red above and revolve it ab
out the x-axis to generate a solid. " }}{PARA 0 "" 0 "" {TEXT -1 76 " \+
A typical cross section will be a washer. We draw a typical cross sect
ion. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 642 "t1 := tubeplot([x
,0,0], x=-1..1, radius=sec(x), axes=normal, style=wireframe, numpoints
=25, tubepoints=80, axes=normal, color=blue):\nt2 := tubeplot([x,0,0],
 x=-1..1, radius=1/2, axes=normal, style=patchnogrid, numpoints=15, tu
bepoints=30, axes=normal, color=brown):\nplane1 := implicitplot3d(\{x=
-1, x=1\}, x=-1..1, y=-2..2, z=-2..2, color=grey, style=patchnogrid):
\nplane1 := implicitplot3d(x=-1, x=-1..1, y=-2..2, z=-2..2, color=grey
, style=wireframe, numpoints=100):\nplane2 := implicitplot3d(x=1, x=-2
..2, y=-2..2, z=-2..2, color=grey, style=wireframe, numpoints=100):\nd
isplay(t1,t2, plane1, plane2, orientation=[-70,70], scaling=constraine
d);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6*-%%MESHG6
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