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I am interested in finding the average Euclidean distance from a point $(x,y)\in\mathbb{D}_2$, the unit disk $\{(u,v):u^2+v^2\leq 1\}\subseteq\mathbb{R}^2$, to the disk $\mathbb{D}_2$. This amounts to essentially calculating the integral $\iint_{\mathbb{D}_2} \|(x,y)-\omega\|_2d\omega.$ By rotational symmetry this gives an integral of the form $\iint_{\mathbb{D}_2}\|(a,0)-\omega\|_2d\omega$ where $0\leq a \leq 1$. I believe this can be converted to an elliptic integral of the second kind where the integral over the angle $\theta$ appears in the form $\int_{0}^{2\pi}\sqrt{1-\frac{4ar}{(r+a)^2}\cos^2(\theta/2)}d\theta.$ I am not sure how to proceed from here.

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    You $m$ight be interested in [this](http://math.stackexchange.com/a/49576), then.2012-08-01

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