I'm evaluating $\iiint_V f(x,y,z) dV$ where V is defined by $2x \le x^2+y^2+z^2 \le 4x $
To simplify things I swapped x and z, and moved to spherical coordinates:
$ 0 \le \theta \le 2\pi, 2 \cos \phi \le \rho \le 4 \cos \phi$
and as the last condition implies $ 4 \cos \phi \ge \rho \ge 0$, I added $0 \le \phi \le \pi/2$
The point is: I have no idea if this parametrization is correct, because I can't figure what that disequations represents. From the spherical parametrization it looks like some paraboloid but I'm confused. Could you help in this? And how do I approach such "confused" situations? thanks.