Evaluating a triple integral Spherical

[PsychonautQQ]

Homework Statement

z(x^2+y^2+z^2)^(-3/2) where x^2+y^2+z^2 ≤ 4 and z ≥ 1

The Attempt at a Solution

So spherically this comes down to cos∅sin∅dpdθd∅
p goes from 0 to 2, theta goes from 0 to 2pi, but I don't know how to figure out what ∅ goes from? I'm trying use trig identities but I'm getting the wrong answer, so maybe this don't work since the sphere is curved?

 

[vela]

Your limits for $\rho$ aren't correct. For example, consider the part of the z-axis which is inside the solid. It's only the part where 1 ≤ z ≤ 2. This would be inconsistent with $\rho \le 1$, yet your lower limit for $\rho$ is 0.

In this case, you have cylindrical symmetry, so try drawing a cross section of the surface through, say, the xz plane. You should be able to see pretty easily the limits of $\phi$, and you'll have to figure out the limits of $\rho$ as a function of $\phi$.

 

[haruspex]

Given the bounds, it might be easier to handle in cylindrical.