Answer
$36 \pi$
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 $div F=\dfrac{\partial A}{\partial x}+\dfrac{\partial B}{\partial y}+\dfrac{\partial C}{\partial z}$
and $div F=\dfrac{\partial (x^2)}{\partial x}+\dfrac{\partial (-y)}{\partial y}+\dfrac{\partial (z)}{\partial z}=2x-1+1=2x$
Consider $I=\iint_{y^2+z^2 \leq 9} \int_0^2 2xdxdydz=\iint_{y^2+z^2 \leq 9} [x^2]_0^2 dydz$
Now, we have
$I=\iint_{y^2+z^2 \leq 9} (4) dydz=4 \cdot \iint_{y^2+z^2 \leq 9} dydz=4 \cdot 9=36 \pi$
