[SOLVED] Calculate a volume + theoretical question

1)Identify the region of integration and calculate the following integral using spherical coordinates : $\displaystyle \int_{-3}^{3} \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} \int _0^{\sqrt{9-x^2-y^2}} z\sqrt{x^2+y^2+z^2} dzdydx$.

My attempt : I notice that the integrand is not the Jacobian of the change of variable spherical to Cartesian but almost. I mean that $\displaystyle z\sqrt{x^2+y^2+z^2}=\rho ^2 \cos \phi$ instead of $\displaystyle \rho ^2 \sin \phi$. Thus I am at a loss.

I don't know how to answer the question.

2) I'd like to know a general method (i.e. applicable in all cases) that transform a complicated solid (like a cone, a paraboloid, hyperboloid or anything different from a cylinder, cube and sphere) into a cube. I'd appreciate any book reference or an example with the above exercise.

I ask this because even though it might be difficult to find such a transformation, if I success in finding it then its Jacobian shouldn't be a problem to find and so the volume of the complicated solid should be quite easy to solve.