The region enclosed by the cone z2 = x2 + y2 between the planes z = 5 and z = 7
I ask again- what is your difficulty? You say you did not come out with 228.28. Well, good for you- the answer is NOT 228.28! Did you come out with something? Please show what you did.
(I think the problem is far simpler in cylindrical coordinates than spherical coordinates.)