Answer
$\dfrac{9}{2}$
Work Step by Step
In order to verify the Divergence Theorem which is true for for the vector field over the region $E$, we will have to add all the surface integrals and should be make sure that all are equal to such as: $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $
Here, we have $S$ shows a closed surface. The region $E$ is inside that surface.
We have $div F=\dfrac{\partial A}{\partial x}+\dfrac{\partial B}{\partial y}+\dfrac{\partial C}{\partial z}$
$I=\int_0^1\int_0^1\int_0^1 (3x+3) dz dy dx =\int_0^1\int_0^1 [3xz+3z]_0^1 dy dx =\int_0^1\int_0^1 [3x(1)+3(1)-0] dy dx $
$=\int_0^1\int_0^1 (3x+3) dy dx $
$=\int_0^1 (3x+3)dx $
$=[(\dfrac{3}{2})x^2+3x]_0^1$
$=(\dfrac{3}{2})(1^2-0)+3(1-0)$
Thus, we have
$=\dfrac{9}{2}$
