I had to cut off the title for the character restriction, here is the full question:
Find the outward flux of the field $\vec{F}(x,y,z) = \langle \frac{x^3}{3}+e^{y^2}, \frac{y^3}{3}+\sin{z}, \frac{z^3}{3}+\pi\cos{\pi x} \rangle$ across the boundary surface of the solid bounded above by $x^2+y^2+z^2=16$ and bounded below by $z=\sqrt{x^2+y^2}$
I know I need to use $\int\int_S \vec{F} \cdot d\vec{S}$ by converting the surface to be a parameterized surface in the $uv$-plane, then use $\int\int_D \vec{F}(\vec{r}(u,v))\cdot(\vec{r}_u \times\vec{r}_v)dA$ where $D$ is the region in the $uv$-plane that I am integrating across, but I am confused on how to find $\vec{r}(u,v)$ to represent the surface.
By the divergence theorem the desired flux is equal to $$\iiint_V (x^2+y^2+z^2) dxdydz=\int_{r=0}^4\int_{\theta=0}^{2\pi}\int_{\phi=0}^{\phi_0} r^2 \cdot r^2\sin(\phi) drd\phi d\theta$$ where $V$ is the solid given by the intersection of the ball and the cone. In the last integral I used the spherical coordinates. Find $\phi_0$ and then integrate.