Double Integral Over Region Common to Two Circles Evaluate $\iint\frac{(x^2+y^2)^2}{x^2y^2} dx dy$ over the region common to the circles $x^2+y^2=7x$ and $x^2+y^2=11y$.

You may use the Cylindrical coordinates to find the value of the integrals. This, as @Ron's Caesarian's approach, will get a bit messy but it works as well. We have two circles on $z=0$ intersected each other, so $$7r\cos\theta=7x=x^2+y^2=11y=11r\sin\theta\longrightarrow\theta_{0}=\tan^{-1}\left(\frac{7}{11}\right)\cong0.56$$ Now we have $$I=\int_{\theta=0}^{\theta_0}\int_{r=0}^{11\sin\theta}\frac{r^4}{r^4\cos^2\theta\sin^2\theta}rdrd\theta+\int_{\theta=\theta_0}^{\pi/2}\int_{r=7\cos\theta}^{0}\frac{r^4}{r^4\cos^2\theta\sin^2\theta}rdrd\theta$$

