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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 17 "Crossed Cylinders" }}
{PARA 19 "" 0 "" {TEXT -1 13 "J. A. Ziegler" }}{PARA 256 "" 0 "" 
{TEXT -1 13 "March 4, 2000" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 256 292 "Finding the volume enclosed by a pair of cylinders of t
he same radius whose axes are perpendicular and in the same plane is a
 well-known exercise in multiple integration.  The difficulty of visua
lizing this object when considered for the first time is also well-kno
wn.  Maple is a great help." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "with(plots):with(plottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }{TEXT 257 60 "We describe the cylinders parametrically and displ
ay them.  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "A:=plot3d([x,
cos(t),sin(t)],x=-1..1,t=0..2*Pi,color=red):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 59 "B:=plot3d([cos(t),y,sin(t)],y=-1..1,t=0..2*Pi,co
lor=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "display(A,B,
scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
258 131 "Now we describe the region common to both and display that.  \+
Notice that a parametric description of the surfaces is not necessary.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "a:=plot3d(\{sqrt(1-x^2)
,-sqrt(1-x^2)\},x=-1..1,y=-x..x,color=red):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 66 "b:=plot3d(\{sqrt(1-y^2),-sqrt(1-y^2)\},x=-y..y,y=-
1..1,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "displ
ay(a,b,scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}{TEXT 259 34 "Next, we use the methods described" }{TEXT -1 1 " " }
{TEXT 260 168 "in the Valdosta Talk (paragraph: Volumes with Vertical \+
Sides) to draw a wedge shaped region containing exactly 1/4 of the vol
ume of the region common to both cylinders." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 64 "s1:=plot3d(\{sqrt(1-x^2),-sqrt(1-x^2)\},x=0..1,y=-
x..x,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "s2:=plo
t3d([x,x,z],x=0..1,z=-sqrt(1-x^2)..sqrt(1-x^2),color=cyan):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "s3:=plot3d([x,-x,z],x=0..1,z
=-sqrt(1-x^2)..sqrt(1-x^2),color=green):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "display(s1,s2,s3,scaling=constrained,view=[0..1.5,-1.
.1,-1..1],axes=normal,style=patchnogrid);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 261 158 "The volume of that part of this which li
es above the x,y-plane is easy to calculate as an iterated integral.  \+
This is 1/8 th of the whole and may be drawn via" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 49 "s4:=plot3d(sqrt(1-x^2),x=0..1,y=-x..x,color=r
ed):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "s5:=plot3d([x,x,z],
x=0..1,z=0..sqrt(1-x^2),color=cyan):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "s6:=plot3d([x,-x,z],x=0..1,z=0..sqrt(1-x^2),color=gre
en):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "display(s4,s5,s6,sc
aling=constrained,view=[0..1.5,-1..1,-1..1],axes=normal,style=patchnog
rid);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 38 "The volu
me of this region is given by " }{XPPEDIT 18 0 "Int(Int(sqrt(1-x^2),y \+
= -x .. x),x = 0 .. 1);" "6#-%$IntG6$-F$6$-%%sqrtG6#,&\"\"\"\"\"\"*$%
\"xG\"\"#!\"\"/%\"yG;,$F/F1F//F/;\"\"!\"\"\"" }{TEXT -1 3 " . " }
{TEXT 263 25 "Calculating this, we find" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "v:=int(int(sqrt(1-x^2),y=-x..x),x=0..1);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 264 23 "So the total volume is " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "V:=8*v;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{MARK "21 0 1" 2 }{VIEWOPTS 1 1 0 1 1 1803 }