Yes, I agree that, because of the circular symmetry, cylindrical coordinates should simplify the integration. I assume that by "the solid bounded by" the cylinder and cone, they mean the region between the two "nappes" of the cone, inside the cylinder.

The easy part- will range from to . Slightly harder: Since the cone, in cylindrical coordinates, is and the cylinder by ( and : and we can cancel an "r"), the cone and cylinder intersect when . For each [itex]\theta[/itex], z ranges from to . Finally, for each z and , r ranges from the cone out to the cylinder. The cone is r= |z| (r is positive, of course) and the cylinder as before.

Your density function is and the differential of volume is .

Putting all of that together, your integral is