To evaluate a surface integral

Asked Apr 27 '20 at 05:04

Active Apr 27 '20 at 05:29

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I was stuck in a question as follows:

$$\int_0^R\int_0^R\frac{(R^2-x^2)(R^2-y^2)}{(x+y)^2}dxdy$$

I tried a lot to simply this but I was unable to do so. I am encountering surface integrals for the first time. I can't make use of the symmetry in $x$ and $y$. I've run out of ideas. Would someone please help me to solve this question?

edited Apr 27 '20 at 05:29

asked Apr 27 '20 at 05:04

shsh23

1,135

1

Perhaps factor the numerator as $(R-x)(R+x)(R-y)(R+y)$ and notice that $x+y=(R+x)-(R-y)=(R+y)-(R-x)$. Disclaimer: I haven't checked that it would help, though. – Alexander Burstein Apr 27 '20 at 05:17
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Change to polar coordinates. – Allawonder Apr 27 '20 at 05:18
Mathematica says it does not converge. – Alexander Burstein Apr 27 '20 at 05:39
Since you say this is a surface integral, could you show us the original problem? Perhaps there was some mistake on your parameterization or some symmetry we can use for the original integral. – Ninad Munshi Apr 27 '20 at 05:43
The original problem was to find the force of a uniform sphere due to one hemisphere on another. – shsh23 Apr 27 '20 at 08:32

To evaluate a surface integral

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