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Can anyone help evaluate

$$ \int_{0}^{r}\int_{0}^{r+\epsilon-\sqrt{r^{2}-x^{2}}}e^{-\beta\left(x^{2}+y^{2}\right)} \, dy \, dx $$

($r$ and $\epsilon$ are constants)?

Michael Hardy
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apg
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    I feel that you should use polar co-ordinates. – Chinny84 Feb 12 '14 at 00:21
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    so just replace x = rcos(theta) and y = rsin(theta) and then try it? (plus rdrd(theta)) – apg Feb 12 '14 at 01:08
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    Yes. But you have to transform the limits, which would lead to integrals of non elementary functions. But give it a go and post your workings for others to see. – Chinny84 Feb 12 '14 at 01:18
  • I don't think this is going to work. You get a tricky Gaussian similar to my question http://math.stackexchange.com/questions/459107/difficult-gaussian-integral-involving-two-trig-functions-in-the-exponent-any-he. Any other ideas? – apg Feb 12 '14 at 19:05
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    I will have a look. But if no one has posted an answer yet it suggests it is very tricky indeed. Is there a need to get an analytical answer? If not use numerical integration? – Chinny84 Feb 13 '14 at 15:12
  • Need an analytic answer... – apg Feb 13 '14 at 18:28

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