I'm having some trouble finding the area under and above the xy-plane
starting off and
so the paraboloid intersects the xy-plane and Mr F says: No. This implies that the region of integration in the xy-plane is a square and, as we both know, it's not. It's a disk (see below).
taking the integral would wield
(this is the part I'm not sure off) coveting to polar form yields
See the correction (in red). The area of the xy-plane you're integrating over is the disk . The radius of this disk is 3, not .