Mind you, I don't actually know how to do this integral directly, but if we use your substitution your integral is

For the y integral, take and recall that for this integration, x is simply a constant. So which means your integral is

Given the result I have a feeling this is not the best way to attack this problem.

I would suggest, perhaps adjusting your variables so that you are integrating over the unit circle rather than an ellipse by using

and

Then your integral becomes:

where the quantity in ( ) is the Jacobian determinant. This is probably more along the lines of what you are looking for.

-Dan