Your problem is that does NOT go from 0 to 1. (And that is why cylindrical coordinates is a much better choice than spherical coordinates). In spherical coordinates, is the straight line distance from (0,0,0) to (x,y,z) which is, of course, [tex]\sqrt{x^2+ y^2+ z^2}[tex]. For any point on the parabola , which is, if my algebra is correct, . In spherical coordinates, and so that becomes .

Solve that equation (quadratic in ; since is never negative, you can discard one root) for and use that as the upper limit of the integral.

I said cylindrical coordinates was a better choice! You have plenty ofcircularsymmetry but nosphericalsymmetry.