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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 52 "Volume, Multiple Integra
tion, and Iterated Integrals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 111 "The cell below illustrates how Maple can be us
ed to draw sets like the subsets of R^2 described in this module." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "The cell
 begins by defining a function  f(x, y)  that is zero if the point (x,
 y) is in the set in question and one otherwise.  Then the procedure  \+
" }{TEXT 256 12 "densityplot " }{TEXT -1 150 " is used to sketch a gra
ph of the set.  Notice that the function f(x, y) must be enclosed in s
ingle quotes.  To speed up the plot you can change the  " }{TEXT 257 
4 "grid" }{TEXT -1 34 "  option to smaller numbers like  " }{TEXT 258 
14 "grid = [32, 32" }{TEXT -1 47 "].  To produce a finer plot you can \+
change the " }{TEXT 259 4 "grid" }{TEXT -1 27 " option to larger numbe
rs.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "with('plots'):\n
\nf := proc(x, y)\nif (-2 <= x) and (x <= 2) and -sqrt(4 - x^2) <= y a
nd y <= sqrt(4 - x^2) then\n   0;\n  else\n   1;\nfi;\nend:\n\ndensity
plot('f(x, y)', x=-3..3, y=-3..3, grid=[64, 64]);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 250 "\nThe next cell illustrates how Maple can graph f
unctions of the form  z = f(x, y).  We are often interested in the por
tion of a function that is above the xy-plane.  The first line uses th
e  max  function to define a function that is never negative.\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "f := (x, y) -> max(0, 2 - x
 - y/2):\n\nplot3d(f(x, y), x=0..4, y=0..4, axes=boxed, scaling=constr
ained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "\nThe next cell illus
trates how Maple can be used to estimate a volume like the volume of t
he tetrahedron in this module." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "f := (x, y) -> evalf(max(0,
 2 - x - y/2)):\n\nest := n ->\n       sum(sum(f((i-1)/n, (j-1)/n)/n^2
,i=1..4*n),j=1..4*n):\n\nest(10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The next cell evaluates the first \+
iterated integral.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 65 "f := (x, y) -> 2 - x - y/2:\n\nint(int(f(
x, y),y=0..4-2*x),x=0..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 53 "The next cell evaluates the second iterat
ed integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 66 "f := (x, y) -> 2 - x - y/2:\n\nint(int(f(x, y), \+
x=0..2-y/2),y=0..4);" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }