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-1 27 "Double and triple integrals" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
234 "Examples from the lecture notes:  (with lots of intermediate step
s given, which would be obtained when doing the calculation by hand, a
nd are shown in the notes; to obtain the final answer, only the last c
alculation is really needed)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Surface integral over tr
iangle (projected onto x-y plane), from p.149 of notes" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "int(12*x+3,y);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&,&*&\"#7\"\"\"%\"xGF'F'\"\"$F'F'%\"yGF'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "expand(int(12*x+3,y=0..4-2*x/3));" 
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\"\"#F&!\"\"\"#7F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "int(%
,x);  " }{TEXT -1 72 "The symbol % indicates the result of the previou
sly evaluated expression" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"#B\"
\"\")%\"xG\"\"#F&F&*&#\"\")\"\"$F&*$)F(F-F&F&!\"\"*&\"#7F&F(F&F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "int(int(12*x+3,y=0..4-2*x/3)
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n is needed" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$C$" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "I
ntegral over the base of the tetrahedron in the x-y plane, from p.150 \+
of the notes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "int(2*x^2+3
,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&\"\"#\"\"\")%\"xGF&F'F'
\"\"$F'F'%\"yGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "expand(
int(2*x^2+3,y=0..4-2*x/3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"
\")\"\"\")%\"xG\"\"#F&F&*&#\"\"%\"\"$F&*$)F(F-F&F&!\"\"\"#7F&*&F)F&F(F
&F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(%,x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,**&#\"\")\"\"$\"\"\"*$)%\"xGF'F(F(F(*&#F(F'
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> " 0 "" {MPLTEXT 1 0 41 "-int(int(2*x^2+3,y=0..4-2*x/3),x=0..6);  " }
{TEXT -1 53 "\nOnly this calculation is needed for the final answer" }
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{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Volume integr
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int(6*x+18*z,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"'\"\"\"%\"
xGF&%\"zGF&F&*&\"\"*F&)F(\"\"#F&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "expand(int(6*x+18*z,z=0..2-x/3-y/2));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,**$)%\"xG\"\"#\"\"\"!\"\"\"#OF(*&\"#=F(%\"yGF(F)*
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\"yGF(!\"\"*&\"#OF(F)F(F(*&\"\"*F()F)F'F(F**&#\"\"$\"\"%F(*$)F)F2F(F(F
(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "expand(int(int(6*x+18*
z,z=0..2-x/3-y/2),y=0..4-2*x/3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
(\"#[\"\"\"*&\"\"%F%)%\"xG\"\"#F%!\"\"*&#F'\"\"*F%*$)F)\"\"$F%F%F%" }}
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" {XPPMATH 20 "6#,(*&\"#[\"\"\"%\"xGF&F&*&#\"\"%\"\"$F&*$)F'F+F&F&!\"
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59 "int(int(int(6*x+18*z,z=0..2-x/3-y/2),y=0..4-2*x/3),x=0..6);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$W\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "The iterated \+
integral can be evaluated in a different order giving the same result:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "int(int(int(6*x+18*z,x=
0..6-3*y/2-3*z),y=0..4-2*z),z=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#\"$W\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "2 \+
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