I'm trying to calculate the flux of the vector field
$$F(x,y,z) = (zy^2-2x,\frac{1}{4}yz+z^2,xy+2x^2+2z)$$
exiting from $\Omega = \{(x,y,z) \in \mathbb{R}^3 \vert x^2 + y^2 + z^2 \leq 16, z \leq \sqrt{x^2+y^2}\}$.
This is what I have done so far: I calculated $\mathrm{div}(F) = \frac{1}{4}z$. Then I tried this integral:
$$ \frac{1}{4}\iiint_{0}^{\sqrt{x^2 + y^2}}zdz $$ $$ \frac{1}{4} \int_{0}^{2\pi} \int_{0}^{\sqrt{8}} \rho^3 d\rho d\theta$$
Then, this sums up to $4\pi$. However the result should be 8$\pi$. Is there some problem with my reasoning?
In spherical coordinates: $$ \int_0^{2\pi}\int_{\pi/4}^{\pi} \int_0^{4} \frac{\rho \cos \phi}{4} \rho^2 \sin \phi \;d\rho d\phi d \theta = -\frac{1}{16} \times 2\pi \times 4^3 = -8 \pi $$