Easiest is probably to use polar (or something equivalent):
But, there's no reason why we couldn't let the -coordinates range between , while the coordinates need to go between :
(In short, there are many parametric representations of the same surface.)
Solution: Start by finding the intersection of these curves. We have which gives the ellipse in the plane We can parametrize these with and , so our top and bottom should look like this:
In fact, I've left bottoms off of all of these, but they would not be hard to add if you wished.