Let $\Omega \subset \mathbb{R}^3$ be the solid given by $$\left\{(x,y,z) \in \mathbb{R}^3:x^2+y^2+z^2 \leq 1; x + 2y − z \geq 0\right\}$$ and $N$ the unitary normal vector field to the surface $\partial \Omega$ that points to the exterior of $\partial \Omega$. Consider the vector field $$F(x,y,z)=(2x-x^2y,xy^2,-z).$$
With this in mind, my goal is to calculate $\iint_{\partial \Omega} \langle F,N \rangle \, \mathrm dS$.
Calculating $\operatorname{div}(F)$: $$\operatorname{div}(F) = \langle\nabla, F\rangle=1.$$ Now as $(0,0,0) \in \left\{(x,y,z) \in \mathbb{R}^3 : x+2y-z=0\right\}$, the solid that I'm integrating over is an half-sphere, so I have that $$\iint_{\partial \Omega} \langle F,N \rangle \, \mathrm dS = \iiint_{ \Omega} \langle \nabla,F \rangle \, \mathrm dV=\frac{2\pi}{3}.$$ Is this a valid argument?