you can try finding the volume of the sphere using polar coordinates/spherical coordinates or the volume of sphere formula if it is allowed, and then subtract it with the volume inside of the cylinder bounded by the sphere.
the volume inside of the cylinder bounded top and bottom by the sphere can be found using
these substitutions x= rcos y=rsin and z=z.
then you should be able to get the answer
(remember to use the appropriate jacobian for the integrand when using these transformations!)
hope this helps