How to convert this specific integral from cartesian to polar

Question

How to convert this surface integral from $x$ and $y$ coordinates to $r$ and $\theta.$ $$\int_0^a\int_0^a\frac{1}{(x^2+y^2+a^2)^{3/2}}dx dy$$ This integral is to get electric field due to a plane from a point charge.

Answer 1

Notice that $dxdy$ is the area of an infinitesimal strip of length and width $dx$ and $dy$ respectively. Now think about an infinitesimal strip of length $dr$ at a distance $r$ from origin and angular width $d\theta$. What would be area? It should be $rdrd\theta$. Thus we would replace $dxdy$ in the integral by $rdrd\theta$ and $x=r\cos \theta$, $y=rsin \theta$ and integrate it over the given square region. We will partition the square region into two parts, the one which lies below and diagonal and the one which lies above the diagonal and calculate the integral as the sum of integrals over these two regions. But the function is symmetrical about the diagonal $x=y$ and hence the two integrals would be same. Thus

\begin{eqnarray} \int_0^a\int_0^a\frac{1}{(x^2+y^2+a^2)^{3/2}}=2\int_0^{\pi/4}\int_0^{\frac{a}{\cos\theta}}\frac{1}{(r^2+a^2)^{3/2}}rdrd\theta \end{eqnarray}