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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "####################
###########################" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "# So
me experience about doing double integral #" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "###############################################" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
50 "restart; E:=0.05; phi_0 := 0.314; psi_0 := 0.349; " }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"EG$\"\"&!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%&phi_0G$\"$9$!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&psi_0G$\"$
\\$!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "term := theta-> \+
1/(1+E/0.511*(1-cos(theta)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%te
rmG:6#%&thetaG6\"6$%)operatorG%&arrowGF(*$,&\"\"\"F.*&%\"EGF.,&F.F.-%$
cosG6#9$!\"\"F.$\"+ir%p&>!\"*F6F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "integrand :=  term(theta)^4 + term(theta)^2 - sin(the
ta)^2*term(theta)^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*integrandG,
(*$,&$\"+et%y4\"!\"*\"\"\"-%$cosG6#%&thetaG$!+5et%y*!#6!\"%F+*$F'!\"#F
+*&-%$sinGF.\"\"#F'!\"$!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 68 "evalf (Int(integrand*(theta-phi_0-psi_0), theta=(phi_0+psi_0)..P
i));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+EeQ_I!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "try := theta-> 1/(2-cos(theta))^2;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tryG:6#%&thetaG6\"6$%)operatorG
%&arrowGF(*$,&\"\"#\"\"\"-%$cosG6#9$!\"\"!\"#F(F(" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 57 "# \"exact answer\" by transforming it to a s
ingle integral:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "evalf (I
nt(try(theta)*(theta-phi_0-psi_0), theta=(phi_0+psi_0)..Pi));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)e*z7_!#5" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 39 "# explicit double integral using \"Int\":" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "evalf (Int(Int(try(phi+psi), phi=ph
i_0..Pi-psi), psi=psi_0..Pi-phi_0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#$\"+)e*z7_!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "\"Int\" \+
is stable if we take Pi = 3.15:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "
evalf (Int(Int(try(phi+psi), phi=phi_0..3.15-psi), psi=psi_0..3.15-phi
_0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+.G*fB&!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "# Now use \"int\", take Pi = 3.14" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "evalf (int(int(try(phi+ps
i), phi=phi_0..3.14-psi), psi=psi_0..3.14-phi_0));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"+V[T3_!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "# Now use \"int\", take Pi = 3.15" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 75 "evalf (int(int(try(phi+psi), phi=phi_0..3.15-psi), psi=psi_0..
3.15-phi_0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+#4F1Z&!\"*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "# => int is unstable, especi
ally when the argument is above Pi. Try the following one:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "evalf (int(int(try(phi+psi), phi=ph
i_0..Pi-psi), psi=psi_0..Pi-phi_0));" }}{PARA 7 "" 1 "" {TEXT -1 32 "W
arning, computation interrupted" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 87 "# It may even spits out imaginary parts sometime. So, if we de
cide to do a complicated " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
67 "# multiple integral numerically, better use \"Int\" instead of \"i
nt\"." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 29 "# Now turn to the real stuff:" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 14 "evalf(Pi, 20);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"5&QKz*e`EfTJ!#>" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 125 "evalf (Int(Int(term(phi+psi)^4 + term(phi+psi)^2 - s
in(phi+psi)^2*term(phi+psi)^3, phi=phi_0..Pi-psi), psi=psi_0..Pi-phi_0
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+EeQ_I!\"*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 129 "evalf (Int(Int(term(phi+psi)^4 + term(ph
i+psi)^2 - sin(phi+psi)^2*term(phi+psi)^3, phi=phi_0..3.14-psi), psi=p
si_0..3.14-phi_0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+J\\pZI!\"*
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "evalf (int(int(term(ph
i+psi)^4 + term(phi+psi)^2 - sin(phi+psi)^2*term(phi+psi)^3, phi=phi_0
..3.14-psi), psi=psi_0..3.14-phi_0));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#$\"+M\\pZI!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "evalf
 (int(int(term(phi+psi)^4 + term(phi+psi)^2 - sin(phi+psi)^2*term(phi+
psi)^3, phi=phi_0..3.15-psi), psi=psi_0..3.15-phi_0));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$!+9O#*>:!\")" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "# See?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "#
 Let me demonstrate more rubbish:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "EP := 1/(2-cos(phi+psi));\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#EPG*$,&\"\"#\"\"\"-%$cosG6#,&%$phiGF(%$psiGF(!\"\"F/
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "evalf(int (int(1/(2-cos
(phi+psi)), phi=0..3.14-psi),  psi=0..3.14));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&$\"+\"4B9O\"!\")\"\"\"%\"IG$!\"%!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "evalf (int(1/(2-cos(theta))*theta, \+
theta=0..3.14));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-4dBA!\"*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "# Who is correct?" }}}}
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