Answer
$8 \pi$
Work Step by Step
Divergence Theorem $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $
Here, $div F=\dfrac{\partial a}{\partial x}+\dfrac{\partial b}{\partial y}+\dfrac{\partial c}{\partial z}$
This implies that $div F=\dfrac{\partial (x^2)}{\partial x}+\dfrac{\partial (xy)}{\partial y}+\dfrac{\partial (z)}{\partial z}=2x+x+1=3x+1$
$I=\int_0^{2 \pi}\int_0^2\int_0^{4-r^2} (3r \cos \theta+1) r dz dr d\theta=\int_0^{2 \pi}\int_0^2\int_0^{4-r^2} [3r^2 \cos \theta+ r] dz dr d\theta$
and $I=\int_0^{2 \pi}\int_0^2 3(4r^2-r^4) \cos \theta+(4r-r^3) dr d\theta=\int_0^{2 \pi}[3(\dfrac{4r^3}{3}-\dfrac{r^5}{5}) \times \cos (\theta)+(2r^2-\dfrac{r^4}{4})]_0^2 d\theta $
Hence, we have $I=\int_0^{2 \pi} 3(\dfrac{32}{3}-\dfrac{32}{5}) \times (\cos \theta)+(8-\dfrac{16}{4}) d\theta=8 \pi$
