The cone does not have "spherical symmetry" so I would be inclined to agree that cylindrical coordinates is better than spherical coordinates. If you are looking for the volume of the region bounded by the sphere and cone, then your limits of integration of should be from to , NOT to 0 (in spherical coordinates, the "co-latitude" goes from 0 to ).

In cylindrical coordinates, [tex]\theta[tex] will go from 0 to [tex]2\pi[/itex], r from 0 to and z from to -r.