Q:

Problem with integration of a function on a ball. 

I have to solve this integral $\int_{x^2+y^2+z^2-2x\le0} \frac{x}{x^2+y^2+z^2}dxdydz$.
So, saw the denominator of the funcion I try substitution with polar ball coordinates, so the integral becomes:
$\int_0^{2\pi}\int_0^\pi\int_0^1\frac{\rho\sin\psi cos\theta}{\rho^2}\rho^2sin\psi d\theta d\psi d\rho$
but the result of this integral is 0, when i know that the result of the first integral is $\pi$.
Can you tell me where I'm wrong. 

A:

$x^2+y^2+z^2-2x\le0$ so $0\le \rho \le 2cos(\theta)sin(\phi)$

Note that $-\frac {\pi}{2} \le \theta \le \frac{\pi}{2}$ 

