Answer
$\dfrac{9 \pi}{2}$
Work Step by Step
Divergence Theorem: $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $
$div F=\dfrac{\partial p}{\partial x}+\dfrac{\partial q}{\partial y}+\dfrac{\partial r}{\partial z}=\dfrac{\partial (3xy^2)}{\partial x}+\dfrac{\partial (xe^z)}{\partial y}+\dfrac{\partial (z^3)}{\partial z}=3y^2+0+3z^2=3(y^2+z^2)$
Now, $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV=\iiint_E 3(y^2+z^2)dV$
or, $=\int_{-1}^{2}\int_0^{2 \pi} \int_0^{1} 3 (r^2 \cos^2 \theta+r^2 \sin^2 \theta) \times [r dr d\theta dx]$
or, $=[\int_{-1}^{2} dx][\int_0^{2 \pi} d\theta][\int_0^{1} 3 r^3 dr]$
or, $=[x]_{-1}^{2} \times [\theta]_0^{2 \pi} \times [3r^4/4]_0^{1} $
$=[2+1] \times [2 \pi-0]_0^{2 \pi} \times [\dfrac{3(1)^4}{4}-0] $
Hence, $I=\dfrac{9 \pi}{2}$