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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 6 "C3M12a" }}}{EXCHG {PARA 
257 "" 0 "" {TEXT -1 50 "Evaluating Integrals using Cylindrical Coordi
nates" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 525 "Let\222s begin with an \+
integral in rectangular coordinates, plot the solid which is the domai
n of the integral, plot the same solid using cylindrical coordinates, \+
and then transform the integral into one using cylindrical coordinates
.  A simple first step is to include the variable of integration in th
e limits of the integral, which allows one to see the appropriate equa
tions that define the surfaces.  Immediately below the integral we wil
l list the necessary packages and the Maple command that would produce
 this integral." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 10 "Example: \001
" }{XPPEDIT 18 0 "Int(Int(Int(x*z,z=0..sqrt(4-x^2-y^2)),x=0..sqrt(1-y^
2)),y=0..1)=Int(Int(Int(x*z,z=(z=0)..(z=sqrt(4-x^2-y^2))),x=(x=0)..(x=
sqrt(1-y^2))),y=(y=0)..(y=1))" "6#/-%$IntG6$-F%6$-F%6$*&%\"xG\"\"\"%\"
zGF-/F.;\"\"!-%%sqrtG6#,(\"\"%F-*$F,\"\"#!\"\"*$%\"yGF8F9/F,;F1-F36#,&
F-F-*$F;F8F9/F;;F1F--F%6$-F%6$-F%6$*&F,F-F.F-/F.;/F.F1/F.-F36#,(F6F-*$
F,F8F9*$F;F8F9/F,;/F,F1/F,-F36#,&F-F-*$F;F8F9/F;;/F;F1/F;F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(student): with(plots):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "A:=Tripleint(x*z,z=0..s
qrt(4-x^2-y^2),x=0..sqrt(1-y^2),y=0..1);\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "valA:=value(A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 39 "We will use plot3d in parametric form, " }{TEXT 257 7 "[x,y,z]
" }{TEXT -1 209 ", with one of the variables eliminated so that just t
wo are allowed to vary and that variable does not appear. The inner in
tegral varies between two surfaces which, for intuitive reasons, we wi
ll designate as " }{TEXT 258 5 "floor" }{TEXT -1 6 "1 and " }{TEXT 
259 4 "roof" }{TEXT -1 107 "1.  Pay close attention to the similaritie
s between the syntax for the triple integral and for these plots." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "floor1:=plot3d([x,y,0],x=0..
sqrt(1-y^2),y=0..1,color=green):\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "roof1:=plot3d([x,y,sqrt(4-x^2-y^2)],x=0..sqrt(1-y^2),
y=0..1,color=blue):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "di
splay(floor1,roof1);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Now let
\222s add the cylindrical wall, " }{TEXT 260 5 "cwall" }{TEXT -1 35 "1
, and the two flat vertical walls." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 64 "cwall1:=plot3d([sqrt(1-y^2),y,z],z=0..sqrt(3),y=0..1,
color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "vwall1:=plot
3d([0,y,z],z=0..sqrt(4-y^2),y=0..1,color=cyan):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 59 "vwall2:=plot3d([x,0,z],z=0..sqrt(4-x^2),x=0..1
,color=plum):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "display(fl
oor1,roof1,cwall1,vwall1,vwall2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
191 "We must see how to sketch this same solid in cylindrical coordina
tes.  We know at this point that the solid is a quarter of a circular \+
cylinder, cut off on the bottom by a horizontal plane ( " }{XPPEDIT 
18 0 "z=0" "6#/%\"zG\"\"!" }{TEXT -1 32 " ) and on the top by a sphere
 ( " }{XPPEDIT 18 0 "x^2+y^2+z^2=4" "6#/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'F
(*$%\"zGF'F(\"\"%" }{TEXT -1 65 " ).  All of this takes place in the f
irst octant which is where  " }{XPPEDIT 18 0 "x>=0" "6#1\"\"!%\"xG" }
{TEXT -1 8 "  and   " }{XPPEDIT 18 0 "y>=0" "6#1\"\"!%\"yG" }{TEXT -1 
162 " .  You will find that this solid was discussed in the section on
 plotting using cylinderplot.  We will use cylinderplot, which is a 3d
-plotter using coordinates " }{XPPEDIT 18 0 "[r,theta,z]" "6#7%%\"rG%&
thetaG%\"zG" }{TEXT -1 47 "   in parametric form.  It may also be used
 as " }{TEXT 261 41 " cylinderplot(f(r,t),r=g(t)..h(t),t=a..b)" }
{TEXT -1 35 "  where the surface is defined by  " }{XPPEDIT 18 0 "z=f(
r,t)" "6#/%\"zG-%\"fG6$%\"rG%\"tG" }{TEXT -1 68 "  in cylindrical coor
dinates.  It is easy to see that we have used  " }{TEXT 262 1 "t" }
{TEXT -1 14 "  instead of  " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }
{TEXT -1 151 "  here.  The figure below has been rotated to provide a \+
better viewing angle.  In cylindrical coordinates we have the followin
g with its output below.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
88 "roof2:=cylinderplot([r,t,sqrt(4-r^2)],r=0..1,t=0..Pi/2,orientation
=[-26,69],color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "f
loor2:=cylinderplot([r,t,0],r=0..1,t=0..Pi/2,color=green):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "cwall2:=cylinderplot([1,t,z],t=0..P
i/2,z=0..sqrt(3),color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 29 "display(floor2,roof2,cwall2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 72 "Adding two more walls gives a figure as the output of the next \+
commands." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "vwall3:=cylind
erplot([r,Pi/2,z],z=0..sqrt(4-r^2),r=0..1,color=plum):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "vwall4:=cylinderplot([r,0,z],z=0..s
qrt(4-r^2),r=0..1,color=cyan):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 43 "display(floor2,roof2,cwall2,vwall3,vwall4);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 214 "We are almost ready to set up our triple integ
ral but first must determine the integrand.  This is the one part of t
he process that is dealt with as a simple direct substitution.  In cyl
indrical coordinates we have" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 2 "
  " }{XPPEDIT 18 0 "x=r*cos(theta)" "6#/%\"xG*&%\"rG\"\"\"-%$cosG6#%&t
hetaGF'" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "y
=r*sin(theta)" "6#/%\"yG*&%\"rG\"\"\"-%$sinG6#%&thetaGF'" }}}{EXCHG 
{PARA 260 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "z=z" "6#/%\"zGF$" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Our integrand in rectangular coor
dinates is   " }{XPPEDIT 18 0 "f(x,y,z)=x*z" "6#/-%\"fG6%%\"xG%\"yG%\"
zG*&F'\"\"\"F)F+" }{TEXT -1 16 "  ,  so in Maple" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 16 "f:=(x,y,z)->x*z;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 36 "F:=simplify(f(r*cos(t),r*sin(t),z));" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 263 10 "IMPORTANT:" }{TEXT -1 19 "  Note the use
 of  " }{TEXT 264 3 "F*r" }{TEXT -1 70 "  for the integrand in our cyl
indical coordinate integral!  Note that " }{TEXT 266 3 " f " }{TEXT 
-1 19 " is a function and " }{TEXT 267 2 " F" }{TEXT -1 19 "  is an ex
pression." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "B:=Tripleint(F
*r,z=0..sqrt(4-r^2),r=0..1,t=0..Pi/2);\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "Bval:=value(B);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
40 "With the limits displayed fully, we have" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 2 "  " }{XPPEDIT 18 0 "Int(Int(Int(x*z,z=(z=0)..(z=sqrt(x^2
-y^2))),x=(x=0)..(x=sqrt(1-y^2))),y=(y=1)..(y=2))=Int(Int(Int(r^2*cos(
theta)*z,z=(z=0)..(z=sqrt(4-r^2))),r=(r=0)..(r=1)),theta=(theta=0)..(t
heta=Pi/2))" "6#/-%$IntG6$-F%6$-F%6$*&%\"xG\"\"\"%\"zGF-/F.;/F.\"\"!/F
.-%%sqrtG6#,&*$F,\"\"#F-*$%\"yGF9!\"\"/F,;/F,F2/F,-F56#,&F-F-*$F;F9F</
F;;/F;F-/F;F9-F%6$-F%6$-F%6$*(%\"rGF9-%$cosG6#%&thetaGF-F.F-/F.;/F.F2/
F.-F56#,&\"\"%F-*$FPF9F</FP;/FPF2/FPF-/FT;/FTF2/FT*&%#PiGF-F9F<" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 265 15 "C3M12a Problems" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 162 "Use Maple to find the value of an equivalent i
ntegral in cylindrical coordinates.  Use cylinderplot to plot the soli
d which represents the domain of the integral." }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 4 "1.  " }{XPPEDIT 18 0 "Int(Int(Int(sqrt(x^2+y^2),z=0..
9-x^2-y^2),y=-sqrt(4-x^2)..sqrt(4-x^2)),x=-2..2)" "6#-%$IntG6$-F$6$-F$
6$-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF0F1/%\"zG;\"\"!,(\"\"*F1*$F/F0
!\"\"*$F3F0F;/F3;,$-F+6#,&\"\"%F1*$F/F0F;F;-F+6#,&FCF1*$F/F0F;/F/;,$F0
F;F0" }{TEXT -1 12 "        2.  " }{XPPEDIT 18 0 "Int(Int(Int(sqrt(x^2
+y^2),z=0..x^2+y^2),y=-sqrt(2*x-x^2)..sqrt(2*x-x^2)),x=0..2)" "6#-%$In
tG6$-F$6$-F$6$-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF0F1/%\"zG;\"\"!,&*
$F/F0F1*$F3F0F1/F3;,$-F+6#,&*&F0F1F/F1F1*$F/F0!\"\"FC-F+6#,&*&F0F1F/F1
F1*$F/F0FC/F/;F7F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "3.  Let  " }
{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 67 "  be an instructor assigned
 constant as described in the preface.  " }{XPPEDIT 18 0 "Int(Int(Int(
sqrt(x^2+y^2),z = 0 .. k-x^2-y^2),y = -sqrt(1-x^2) .. sqrt(1-x^2)),x =
 -1 .. 1);" "6#-%$IntG6$-F$6$-F$6$-%%sqrtG6#,&*$)%\"xG\"\"#\"\"\"F2*$)
%\"yGF1F2F2/%\"zG;\"\"!,(%\"kGF2*$F/F2!\"\"*$F4F2F=/F5;,$-F+6#,&F2F2*$
F/F2F=F=-F+6#,&F2F2*$F/F2F=/F0;,$F2F=F2" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}}{MARK "40 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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