# non continuous double integration

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Ahmed Abdulla on 6 Jun 2021
Answered: John D'Errico on 6 Jun 2021
I am looking to integrate the following function 5sin(x) dx dy, however I am only aware of performing integrals for specific upper bounds using integral2.
Does anyone know if it is possible/how to perform this double integration given x and y are non continuous set of numbers. x and y, x=[20 33 47 85] and y=[10 22 43 98].
*Sorry if the title isn't clear
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Ahmed Abdulla on 6 Jun 2021
Thank you for your time, sorry I am not very sure how to word it.
To apply the trapezoidal rule, should i evaluate the integrated function at each set of points then sum up the findings?

John D'Errico on 6 Jun 2021
Ok, now that we understand the question, it is easy. Just use trapz, TWICE.
To apply the trapezoidal rule, you will create a matrix of 4x4 elements. Then call trapz twice, once on each dimension of the array. I won't do your problem here, since this seems too much likely to be homework.
But for example, suppose I wanted to integrate the function
z = x.^3 + x.*y^2
over the domain (x,y) = [20,85] X [10,98], where we will sample the function at that set of nodes? This is VERY different from what you wanted to call a discontinuous point set of some sort.
Zfun = @(X,Y) X.^3 + X.*Y.^2;
Xnodes = [20 33 47 85];
Ynodes = [10 22 43 98];
[x,y] = meshgrid(Xnodes,Ynodes)
x = 4×4
20 33 47 85 20 33 47 85 20 33 47 85 20 33 47 85
y = 4×4
10 10 10 10 22 22 22 22 43 43 43 43 98 98 98 98
z = Zfun(x,y)
z = 4×4
10000 39237 108523 622625 17680 51909 126571 655265 44980 96954 190726 771290 200080 352869 555211 1430465
intest = trapz(Xnodes,trapz(Ynodes,z,1),2)
intest = 2.4820e+09
How accurate is that estimate? Remember, this is just trapezoidal integration, over a VERY coarse set of nodes.
syms X Y
int(int(X.^3 + X.*Y.^2,X,20,85),Y,10,98)
ans =
2214362150
So the trapezoidal estimate was within 10% of the exact value. Probably as good as we could do on that set of nodes.

R2019a

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