Using geometry, calculate the volume of the solid under z=64−x2−y2and over the circular disk x2+y2≤64

Perhaps you mean to find the volume under ? It seems like you have to rely on calculus otherwise (in the case that you indeed mean ).

Assuming this, note that describes the top half of a sphere with radius 8, which means the volume below this surface over the disk is simply half the volume of the sphere. Thus the volume is