Om vi t ex använder den uttömmande följden \(D_n=\{(x,y):x,y\ge 0, x^2+y^2\le n^2\}\), \(n=1,2,3,\ldots \), så kan integralen skrivas \[ I=\lim_{n\to\infty}\iint_{D_n}\frac{dxdy}{1+(x^2+y^2)^2}= \Big[ \begin{array}{cl} x=&r\cos\theta\\ y=&r\sin\theta \end{array} \Big]= \] \[ \lim_{n\to\infty}\int_0^{\pi/2} d\theta \int_0^n\frac{rdr}{1+r^4}= \Big[ \begin{array}{cl} t=&r^2\\ dt=&2rdr \end{array} \Big]= \] \[ \lim_{n\to\infty}\frac{\pi}{2} \int_0^{n^2}\frac{\frac12 dt}{1+t^2}= \frac{\pi}{4}\lim_{n\to\infty}\Big[ \arctan t\Big]_0^{n^2}=\frac{\pi^2}{8}. \]