{VERSION 6 0 "IBM INTEL NT" "6.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 18 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim
es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 
2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 25 "Several Variables, Part
 3" }}{PARA 256 "" 0 "" {TEXT 263 11 "Integration" }}{PARA 256 "" 0 "
" {TEXT -1 10 "April 2009" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta
rt;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Double intergals or triple
 integrals can be worked by nesting the " }{TEXT 259 3 "int" }{TEXT 
-1 8 " or the " }{TEXT 260 3 "Int" }{TEXT -1 12 " command.   " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 7 "Example" 
}{TEXT -1 14 ": Compute     " }{XPPEDIT 18 0 "int(int(int(6*x*y*z,z = \+
y .. 1),y = 0 .. x),x = 0 .. 1);" "6#-%$intG6$-F$6$-F$6$**\"\"'\"\"\"%
\"xGF,%\"yGF,%\"zGF,/F/;F.F,/F.;\"\"!F-/F-;F4F," }{TEXT -1 17 "  .  \n
We enter : " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "int( int( int( 6*x*y
*z, z = y..1), y =0..x),x=0..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
214 "and obtain a result.  The problem with this approach is that you \+
don't see the integral which has been computed, and you need to have f
aith that everything was entered correctly.  As an alternative you can
 use the " }{TEXT 258 14 "Int + value(%)" }{TEXT -1 13 " combination:
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Int( Int( Int( 6*x*y*z, z = y..
1), y =0..x),x=0..1);\nvalue(%);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "We can also \+
compute this integral by working our way from the inside out - the way
 you would do it with pencil and paper - beginning with beginning with
 the innermost integral:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Int( 6*
x*y*z, z = y..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Int( % , y=0..x);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "Int( % , x=0..1);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "It is advisa
ble to make maple's task as easy as possible.  Otherwise it might free
ze up on you, or it may not be able to come up with a closed form solu
tion.\n" }{TEXT 261 7 "Example" }{TEXT -1 14 ":  Compute    " }
{XPPEDIT 18 0 "Int(Int(4*y*sin(x^2),x = y^2 .. sqrt(Pi)),y = 0 .. Pi^(
1/4));" "6#-%$IntG6$-F$6$*(\"\"%\"\"\"%\"yGF*-%$sinG6#*$%\"xG\"\"#F*/F
0;*$F+F1-%%sqrtG6#%#PiG/F+;\"\"!)F8*&F*F*F)!\"\"" }{TEXT -1 105 ".   T
his problem is taken from the practice sheet for the second test.  Dir
ect implementation results in " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "I
nt( Int( 4*y*sin(x^2),x=y^2..sqrt(Pi)) , y = 0..Pi^(1/4));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
108 "Maple cannot find a simple closed form expression for the solutio
n.  Resorting to a numerical answer we find" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "By \+
changing the order of integration (some sketches are required) the sam
e integral can be written as\n" }{MPLTEXT 1 0 53 "Int( Int( 4*y*sin(x^
2),y=0..sqrt(x)), x=0..sqrt(Pi));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
32 "and the integral is found to be " }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "without further
 complications." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "On other occa
sions it is advisable to convert the integral to polar coordinates." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 8 "Example:
" }{TEXT -1 12 "  Compute   " }{XPPEDIT 18 0 "Int(Int(sin(x^2+y^2),y =
 -sqrt(Pi/2-x^2) .. sqrt(Pi/2-x^2)),x = 0 .. sqrt(Pi/2));" "6#-%$IntG6
$-F$6$-%$sinG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF.F//F1;,$-%%sqrtG6#,&*&%#Pi
GF/F.!\"\"F/*$F-F.F;F;-F66#,&*&F:F/F.F;F/*$F-F.F;/F-;\"\"!-F66#*&F:F/F
.F;" }{TEXT -1 5 "  .  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 33 "Direct implenentation results in " }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 80 "Int(Int( sin(x^2+y^2), y=-sqrt(Pi/2-x^2)..sqrt(Pi/
2-x^2)), x = 0 .. sqrt(Pi/2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 9 "value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Maple is stuc
k, and a numerical calculation results in " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "In polar coordinates the int
egral becomes" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Int(Int( r*sin(r^2
),r=0..sqrt(Pi/2)), theta=0..Pi);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
41 "which can be computed without any trouble" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 2 0" 10 
}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }