TITLE: Compute a multiple integral.
QUESTION [5 upvotes]: A friend of mine asked me to help compute the multiple integral:$$\iint_{(D)} \frac{x^2 y^2}{4+9x^4 y^4}\text{d}\sigma, \ \ D:x^2+y^2\le2.$$ But I find it extremely hard. Is there a simpler way to compute it?

REPLY [2 votes]: I think this is going to be long no matter what you do; however, there is a way that will work, with care. Switch to polar coordinates, and remember the $r \, dr \, d\theta$
Now, any rational function in $\cos \theta$ and $\sin \theta$ can be changed to a rational function of a variable $t$ by this trick. Taking $- \pi < \theta < \pi$ and $- \infty < t < \infty,$
$$   \theta = 2 \arctan t $$
$$ \tan \theta = \frac{2t}{1-t^2},  $$
$$ \cos \theta = \frac{1-t^2}{1+t^2},  $$
$$ \sin \theta = \frac{2t}{1+t^2},  $$
$$ d \theta  = \frac{2}{1+t^2} \; dt.  $$
Work is then continued with partial fractions. 
Editttt; wrote it out. I knew this was stereographic projection, I had details wrong in my mind. Take the point $(0,t)$ on the $y$-axis. Connect it to the point $(-1,0)$ by a line. This line intersects the standard unit circle $x^2 + y^2 = 1$ in two points, $(-1,0)$ and the point I am calling $(\cos \theta, \sin \theta).$