Let F be the vector field given by $$\mathbf{F}(x,y,z) = (x^2y + y^2, z - xy^2, 3z^2) $$ and let $T$ be the region in space enclosed by the sphere $x^2 + y^2 +z^2 = 4$, and the $xz$-plane where $x \geq 0$ and the $yz$-plane where $y \geq 0$. Calculate $$\int \int_S \mathbf{F} \cdot \mathbf{\hat{N}}dS$$ where $S$ is the part of the boundary of $T$ lying on the spherical surface $x^2 + y^2 + z^2 = 4$ and with the unit normal vector $\mathbf{\hat{N}}$ pointing away from $T$.
Im interpreting the region as being the part of thes phere where $x,y > 0$ and $ -2 \leq z \leq 2$. Applying the divergence theorem I then get $$\int_0^\pi \int_0^\pi \int_0^26\rho^3sin(\phi)cos(\phi)drd\phi d\theta = 0 $$which is incorrect. What am I missing here?