# How to calculate a double integral inside the domain of intersection of two functions?

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Mehdi il 16 Gen 2023
Modificato: Torsten il 25 Gen 2023
I want to calculate a double integral of an arbitrary function (f) inside the region of intersection of two other functions. Please suggest a fast and convenient approach.
clear
JJ = 5;
II = 5;
W = rand(II, JJ);
syms x y
w = sym('0');
f = sym('0');
for i=1:II
for j=1:JJ
w =w+W(i, j)*legendreP(i-1, x)*legendreP(j-1, y);
f =f+(legendreP(i-1, x)*legendreP(j-1, y))^2;
end
end
H = 0.5*(1+tanh(w));
fsurf(w,[-1,1,-1,1],'red')
hold on
% figure
fsurf(H,[-1,1,-1,1],'blue')
%F = double integral of f inside the domain of intersection of two functions as
%the region showed in pic
##### 13 CommentiMostra 11 commenti meno recentiNascondi 11 commenti meno recenti
Mehdi il 18 Gen 2023
Any suggestion to speed up the calculation?
clc
clear
JJ = 9;
II = 9;
W = 01*rand(II, JJ);
syms x y
w = sym('0');
f = sym('0');
for i=1:II
for j=1:JJ
w =w+W(i, j)*legendreP(i-1, x)*legendreP(j-1, y);
f =f+(legendreP(i-1, x)*legendreP(j-1, y))^2;
end
end
H = 0.5*(1+tanh(11*w));
f = matlabFunction(f,'Vars',[x y]);
g = matlabFunction(H-0.5,'Vars',[x y]);
D = @(x,y)f(x,y).*(g(x,y)>0);
Dint = integral2(D,-1,1,-1,1)
Torsten il 18 Gen 2023
Modificato: Torsten il 18 Gen 2023
Any suggestion to speed up the calculation?
No. I remember we had this integration problem before. And the region where g(x,y)>0 is quite complicated.

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### Risposte (1)

Bjorn Gustavsson il 17 Gen 2023
Modificato: Bjorn Gustavsson il 17 Gen 2023
Perhaps you can use Green's theorem (you'd be very lucky if you could - but if you were to be that lucky in this case it would be a shame to miss it). That would take you from a sum of integrals over rather complicated regions to perhaps simpler integrals around the boundaries of the region. That would be nice. Given the shape of your function it doesn't seem entirely improbable.
For the case where you actually have to perform the calculations you would use the steps suggested in @Torsten's comment.
HTH
##### 5 CommentiMostra 3 commenti meno recentiNascondi 3 commenti meno recenti
Mehdi il 25 Gen 2023
Modificato: Mehdi il 25 Gen 2023
C can be found by FEX submission availiable through https://www.mathworks.com/matlabcentral/fileexchange/74010-getcontourlinecoordinates?s_tid=srchtitle. The problem for me is to find M and L (a,b). Any suggestions?
Torsten il 25 Gen 2023
Modificato: Torsten il 25 Gen 2023
a = b = 0.
You can use any function M with dM/dx = f.
But you need a symbolic integration here.
I doubt the whole process will be faster than using integral2 directly, apart from the problems of evaluating the curve integral.

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