Answer
$\dfrac{9}{2}$
Work Step by Step
$div F=\dfrac{\partial a}{\partial x}+\dfrac{\partial b}{\partial y}+\dfrac{\partial c}{\partial z}$
This implies that $div F=\dfrac{\partial (xye^z)}{\partial x}+\dfrac{\partial (xy^2z^3)}{\partial y}+\dfrac{\partial (-ye^z)}{\partial z}=ye^z+2xyz^3-ye^z=2xyz^3$
The Divergence Theorem states that $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $
$=\int_{0}^3\int_0^2 \int_0^1 (2xyz^3)dzdydx$
$=\int_{0}^3\int_0^2[\dfrac{xy}{2}]dydx$
$=\int_{0}^3 x dx$
$=[x^2/2]_0^3$
$=\dfrac{9}{2}$
