Hi there, I have to compute the surface area for

I have a problem on finding the surface area for the paraboloid limited by the plane. I've parametrized the plane in polar coordinates, I thought it would be easier this way, but also tried in cartesian coordinates with the same result. The problem I have is to set the limits of integration on such a way that determines the area of the paraboloid limited by the plane.

So the paraboloid is parametrized by:

Then

Now the surface area is determined by:

Now D is the region inside the disc determined by the intersection of the surfaces. So D:

And this is my region of integration for my paraboloid. Now it doesn't seem so easy to express the region for the parametrization I choose.

So I'm trying to solve this in cartesian coordinates, this is the integral for the surface in cartesian coord:

But this integral isn't so easy to solve. I tried to go from here to cylindrical coordinates, using the substitution that helps a bit with the limits of integration.

Then I get:

Thats the best expression I get, but still too complicated to integrate by hand.