Using Divergence theorem to calculate flux

The divergence theorem allows us to write $$ \iint_S\vec{F}\cdot d\vec{S}=\iiint_W \nabla\cdot \vec{F}\; dV = \iiint_W 2z\; dV $$ where $W$ is the region bounded by $x^2+y^2=4$, $z=x+1$ and $z=0$, i.e., $$ W=\{(x,y,z)\;|\; -1\le x \le 2,\; 0\le z\le 1+x,\; -\sqrt{4-x^2}\le y \le \sqrt{4-x^2} \} $$ It follows that $$ \iiint_W 2z\; dV = \int_{-1}^2\int_0^{1+x}\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}2z\; dydzdx = \int_{-1}^2\int_0^{1+x} 4z\sqrt{4-x^2}\; dzdx \\= \int_{-1}^22(1+x)^2\sqrt{4-x^2}\; dx = \frac{16\pi}{3}+\frac{9\sqrt{3}}{2} $$