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
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.