Well, the first thing is to sketch the region bounded by the plane and the paraboloid. Then find the intersection, which is
, an ellipse in the plane but whose projection on is a circle , centre and radius . So far, so good.
The volume of the required region is , where and . This is the 3-D equivalent of the formula for the area between two curves in the -plane.
So you need to evaluate and yes, the best way involves a change to polar coordinates.
In this, the element of area becomes and as the region of integration is now . Therefore