This paper http://math.ucsb.edu/~cmart07/Evaluating%20Integrals.pdf hints at a way to compute the sum $ \sum_{n=1}^\infty \frac{1}{n^2} $ by expanding it into the double integral $\int_0^1 \int_0^1 \frac{\mathrm{d}x \, \mathrm{d}y}{1-xy}.$ Now for solving this integral, the paper suggests rotating the area $[0,1]^2$ by $45^\circ$ for then to rewrite it in polar coordinates.
I have made a sketch of the area below, but I am having problems rewriting the integral in polar coordinates.
Dissregarding the function I was thinking the limits would have to be
$\int_{-\pi/4}^{\pi/4} \int_0^{\sqrt{\cos^4\theta + (\sqrt{2}-\cos^2\theta)^2}}r \,\mathrm{d}r\,\mathrm{d}\theta$ but the upper limit is wrong for the radius. Hmm... I was basically finding the distance from the function $f(x) = \sqrt{2}-x$ to origo, then converting this to polar..
Any help computing the sum using the double integral transform would be very appreceated. I already know several methods for computing the bessel identity, however this one stumped me.