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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 22 "Intersecting Cylinders" }}
{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 9 "Jim He
rod" }}{PARA 259 "" 0 "" {TEXT -1 21 "School of Mathematics" }}{PARA
260 "" 0 "" {TEXT -1 12 "Georgia Tech" }}{PARA 261 "" 0 "" {TEXT -1
21 "herod@math.gatech.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 556 " This worksheet is about visualization of reg
ions in three dimensions. Imagine two cylinders of radius 1 which inte
rsect at right angles. Perhaps you can see the solid which would be t
hat intersection. Can you draw it? Can you describe it to a colleague \+
well enough that how to compute the volume of the solid will be transp
arent? In addition to computing that volume here, you should appreciat
e even more how the computer helps in giving visualization to solids. \+
Maybe you will enjoy adding the techniques for drawing these solids to
your repertoire." }}{PARA 0 "" 0 "" {TEXT -1 325 " The problem o
f finding the volume of two intersecting cylinders is one of those cla
ssical problems for three dimensional integration that all students wh
o pass here encounter. We suppose that the cylinders have radius 1: th
e intersections of the cylinders with the x-y plane and the y-z plane \+
are given by the equations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0
"" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$%\"xG\"
\"#\"\"\"*$%\"yGF'F(F(" }{TEXT -1 10 " and " }{XPPEDIT 18 0 "z^2+
y^2=1" "6#/,&*$%\"zG\"\"#\"\"\"*$%\"yGF'F(F(" }{TEXT -1 1 "." }}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "In an attempt \+
to visualize the intersection, we first draw the two cylinders, one re
d and the other blue." }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 72 "J:=cylinderplot(1,theta=0..2*Pi,z=-1..1,color=BLUE,scaling=const
rained):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "K:=plot3d(\{sqr
t(1-y^2),-sqrt(1-y^2)\},y=-1..1,x=-2..2,color=RED):" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 17 "display3d(\{J,K\});" }}}{PARA 0 "" 0 ""
{TEXT -1 264 " Look at that last graph and see if the following th
ree pictures do not give visualization to the intersection. The first \+
picture is from the perspective of standing in the first octant with t
he same orientation as above. We make the blue cylinder transparent."
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "plot3d(\{sqrt(1-y^2),-sqr
t(1-y^2)\},y=-sqrt(1-x^2)..sqrt(1-x^2),x=-1..1,\norientation=[45,45],a
xes=NORMAL,color=RED);" }}}{PARA 0 "" 0 "" {TEXT -1 145 " In the a
bove view you are looking down into the red part of the intersection. \+
Next, we look from the first quadrant, but near to the x axis." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "plot3d(\{sqrt(1-y^2),-sqrt(
1-y^2)\},y=-sqrt(1-x^2)..sqrt(1-x^2),x=-1..1,\norientation=[15,45],axe
s=NORMAL,color=RED);" }}}{PARA 0 "" 0 "" {TEXT -1 94 "Finally, look at
the surface from the fourth quadrant, close to the plane of the x and
y axes." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "plot3d(\{sqrt(1
-y^2),-sqrt(1-y^2)\},y=-sqrt(1-x^2)..sqrt(1-x^2),x=-1..1,\norientation
=[125,85],axes=NORMAL,color=RED);" }}}{PARA 0 "" 0 "" {TEXT -1 75 " \+
We ask: what does the surface look like just above the first quadran
t?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "J:=plot3d(\{sqrt(1-y^
2)\},y=0..sqrt(1-x^2),x=0..1,\norientation=[-35,55],axes=NORMAL,style=
hidden,light=[50,25,0,1,0]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 104 "K:=plot3d(0,y=0..sqrt(1-x^2),x=0..1,\norientation=[-35,55],axes
=NORMAL,style=hidden,light=[50,25,1,0,0]):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 17 "display3d(\{J,K\});" }}}{PARA 0 "" 0 "" {TEXT -1
174 " Now perhaps you see how to compute the volume. Compute the vo
lume of the entire intersection by computing the volume indicated in t
he above picture and multiply by eight." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1
}{PAGENUMBERS 0 1 2 33 1 1 }