The elipse in polar coordinates is , so:
I'm trying to evaluate over the region enclosed by the ellipse .
Transforming the region to a circle of radius 1 (with the transformation ) and then to polar coordinates gives the result , which corresponds to the key of the problem.
However, I figured I would attempt to do the integration again by going directly to polar coordinates, giving, after a bit more work
.
I checked the latter result by integrating the integral after the first equality sign in the row above in Mathematica, giving . Thus, it seems likely that there's some conceptual misunderstanding on my part. What exactly did I do wrong?