Set up the integral needed to find the volume of the solid bounded by the hyperboloid z2 = 64 x2 y2 and by the upper nappe of the cone z2 = 2x2 2y2

It's clear enough that the upper half of the cone falls below the upper sheet of the paraboloid, so that . Right away you can see that converting to cylindrical coordinates will be quite advantageous.

The intersection of the two surfaces occurs as a circle:



which is parallel to the x-y plane, has radius 8, and is centered at .

The volume of this space is given by the integral



where denotes the bounded space between the surfaces. Converting to cylindrical coordinates, this can be expressed as



which evaluates to .