I was tutoring someone and one of the problems was to integrate $r^4\cos^2\theta\sin\theta$ over the portion of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ that lies in the first quadrant. So, $0\leq \theta\leq\frac{\pi}{2}$. But, given a $\theta$, $r$ runs from $0$ to
$ \sqrt{\frac{36+36\tan^2\theta}{9+4\tan^2\theta}}. $
After integrating with respect to $r$ first, it looks very nasty.
Is there some clever method, such as a change of variables, that makes this easy to integrate?