I have done the following triple integral $ \displaystyle \iiint 2x ~dx~dy~dz~$ on the region defined by $S={x^2 +y^2 + z^2 \le 1 }$.
Turning to the spherical coordinates I get
$$ \left\{ \begin{array}{c} x = \rho \cos\theta \sin\phi \\ y= \rho\sin\theta \sin\phi \\ z = \rho\cos\phi \end{array} \right. $$
where $dx~dy~dz = \rho^2\sin\phi ~d\rho~d\theta~d\phi$ and $\rho \in [0,1], \theta \in [0, 2 \pi], \phi \in [0, \pi] $
The integral therefore becomes
$ \displaystyle 2 \int_0^1 \rho^3 ~d\rho \int_0^{2\pi}\cos\theta~ d\theta \int_0^{\pi}\sin^2 \phi ~d\phi ~$ and the result is $0$.
I'm not sure I did the calculations correctly