Answer
$\dfrac{9}{2}$
Work Step by Step
Divergence Theorem: $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $
$div F=\dfrac{\partial a}{\partial x}+\dfrac{\partial b}{\partial y}+\dfrac{\partial c}{\partial z}=ye^z+2xyz^3-ye^z=2xyz^3$
Consider $I=\int_{0}^3\int_0^2 \int_0^1 (2xyz^3) dzdydx==\int_{0}^3\int_0^2[\dfrac{xy}{2}]dydx$
Hence, we get $I=\int_{0}^3 x dx=[\dfrac{x^2}{2}]_0^3$
and $I=\dfrac{3^2}{2}-\dfrac{0^2}{2}=\dfrac{9}{2}$
