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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "read(`c:/othermaple/
plot3d_region02.txt`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "h
elp_plot3d_region();" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
261 80 "For all these, one would want to sketch the bounding surfaces \+
and regions before" }}{PARA 0 "" 0 "" {TEXT 262 82 "trying to set up t
he limits of integration. (The function plot3d_region is really " }}
{PARA 0 "" 0 "" {TEXT 263 68 "visualizing the limits supplied, not fig
uring out the right limits.)" }}{PARA 256 "" 0 "" {TEXT -1 81 "However
, to give you a better feel for what we are seeking in setting up a tr
iple" }}{PARA 256 "" 0 "" {TEXT -1 76 "integral, spend a little time t
hinking at how the solid pictured arises from" }}{PARA 256 "" 0 "" 
{TEXT -1 51 "the problem specification and the limits supplied. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 36 "Remember
 that in the triple integral" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(int(
int(f(x,y,z),z = g(x,y) .. h(x,y)),y = j(x) .. k(x)),x = a .. b);" "6#
-%$intG6$-F$6$-F$6$-%\"fG6%%\"xG%\"yG%\"zG/F/;-%\"gG6$F-F.-%\"hG6$F-F.
/F.;-%\"jG6#F--%\"kG6#F-/F-;%\"aG%\"bG" }{TEXT -1 1 "," }}{PARA 0 "" 
0 "" {TEXT -1 19 "                   " }{TEXT 268 74 "1. For fixed val
ues of x and y, the limits g(x,y) and h(x,y) describe the " }}{PARA 
256 "" 0 "" {TEXT -1 100 "                       intersection with the
 solid of the corresponding line parallel to the z axis." }}{PARA 0 "
" 0 "" {TEXT -1 20 "                    " }{TEXT 269 39 "2. The outer \+
two limits of integration " }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(int(F,
y = j(x) .. k(x)),x = a .. b);" "6#-%$intG6$-F$6$%\"FG/%\"yG;-%\"jG6#%
\"xG-%\"kG6#F//F/;%\"aG%\"bG" }{TEXT -1 1 " " }{TEXT 270 36 "(after th
e inner z integral is done)" }}{PARA 0 "" 0 "" {TEXT -1 21 "          \+
           " }{TEXT 271 58 "   describe the projection of the solid in
to the xy plane." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "Cube. (The S
implest Case.)" }}{EXCHG {PARA 0 "" 0 "" {TEXT 256 39 "Unit Cube 0 <=x
 <=1,0 <=y<=1,0 <=z <=1." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "gra:=p
lot3d_region(x=-1..1,y=-1..1,z=-1..1,11):\ndisplay(gra); display([gra[
1],gra[2]],title=`Top and Bottom`);  display([op(3..6,gra)],title=`Sid
es`);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 71 "Solid with Six Pieces \+
to its Boundary. (The Generic Elementary Region.)" }}{EXCHG {PARA 256 
"" 0 "" {TEXT -1 71 "Region in the first octant bounded above by z=x^2
+y^2+1, below by z=0, " }}{PARA 256 "" 0 "" {TEXT -1 73 "and on the si
des by the xz plane, yz plane, the plane y=2 as well as the " }}{PARA 
256 "" 0 "" {TEXT -1 31 "parabolic cylinder x=(y-1)^2+1." }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 163 "gr0:=plot3d_region(y=0..2,x=0..(y-1)^2+1,z=0
..1+x^2+y^2,20):\ndisplay(gr0); display([gr0[1],gr0[2]],title=`Top and
 Bottom`);  display([op(3..6,gr0)],title=`Sides`);" }}}}{SECT 1 {PARA 
3 "" 0 "" {TEXT -1 12 "Tetrahedron." }}{EXCHG {PARA 0 "" 0 "" {TEXT 
260 53 "Solid Bounded by the planes x=0, y=0, z=0, 2x+y+3z=6." }{TEXT 
-1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "gr5:=plot3d_region(x=0..
3,y=0..6-2*x,z=0..(6-2*x-y)/3,20):\ndisplay(gr5); display([gr5[1],gr5[
2]],title=`Top and Bottom`);  display([op(3..6,gr5)],title=`Sides`);" 
}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Between Two Paraboloids." }}
{EXCHG {PARA 0 "" 0 "" {TEXT 272 63 "Solid Bounded by the paraboloids \+
z=x^2+2y^2 and z=12-2*x^2-y^2." }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 188 "gr6:=plot3d_region(x=-2..2,y=-sqrt(4-x^2)..sqrt(4-x^
2),z=x^2+2*y^2..12-2*x^2-y^2,20):\ndisplay(gr6); display([gr6[1],gr6[2
]],title=`Top and Bottom`);  display([op(3..6,gr6)],title=`Sides`);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 
"" {TEXT -1 28 "Cylinder Cut By Some Planes." }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 259 78 "Solid in the first octant cut off by th
e plane z=y and the cylinder 1=x^2+y^2." }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 157 "gr2:=plot3d_region(x=0..1,y=0..sqrt(1-x^2),z=0..y,20):\ndispl
ay(gr2); display([gr2[1],gr2[2]],title=`Top and Bottom`);  display([op
(3..6,gr2)],title=`Sides`);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "
Solid Bounded by Some Planes." }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 265 56 "Solid Bounded by the planes x=0, \+
y=0, z=0, y+z=1, x+z=1." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 
41 "(Looking first at the graphs helps here.)" }{TEXT -1 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 112 "g2:=plot3d(1-x,x=0..2,y=0..2,color=red):
\ng3:=plot3d(1-y,x=0..2,y=0..2,color=blue):\ndisplay([g2,g3],axes=norm
al);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 99 "Since the top of the sol
id is sometimes z=1-x and sometimes z=1-y, it is natural to use two pi
eces." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 420 "gr3:=plot3d_region(x=0..1
,y=0..x,z=0..1-x,20):\ngr4:=plot3d_region(x=0..1,y=x..1,z=0..1-y,20):
\n\ndisplay([op(gr3),op(gr4)],title=`The Two Elementary Regions Combin
ed`);\ndisplay(gr3); display([gr3[1],gr3[2]],title=`Top and Bottom`); \+
 display([op(3..6,gr3)],title=`Sides`);\n\ngr4:=plot3d_region(x=0..1,y
=x..1,z=0..1-y,20):\ndisplay(gr4); display([gr4[1],gr4[2]],title=`Top \+
and Bottom`);  display([op(3..6,gr4)],title=`Sides`);" }}}{PARA 3 "" 
0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Between Two
 Parabolic Cylinders" }}{EXCHG {PARA 0 "" 0 "" {TEXT 257 55 "The Solid
  Bounded Above by z=1-x^2 and Below by z=y^2." }}{PARA 0 "" 0 "" 
{TEXT 258 41 "(Looking first at the graphs helps here.)" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 118 "g0:=plot3d(1-x^2,x=-2..2,y=-2..2,color=red):
\ng1:=plot3d(y^2,x=-2..2,y=-2..2,color=blue):\ndisplay([g0,g1],axes=no
rmal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "gr1:=plot3d_regi
on(x=-1..1,y=-sqrt(1-x^2)..sqrt(1-x^2),z=y^2..1-x^2,20):\ndisplay(gr1)
; display([gr1[1],gr1[2]],title=`Top and Bottom`);  display([op(3..6,g
r1)],title=`Sides`);" }}}}}{MARK "0 0 0" 19 }{VIEWOPTS 1 1 0 1 1 1803 
1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }