Answer
$36 \pi$
Work Step by Step
Given: $F=\lt x^2, -y,z \gt$
$div F=\dfrac{\partial a}{\partial x}+\dfrac{\partial b}{\partial y}+\dfrac{\partial c}{\partial z}$
This implies that $div F=\dfrac{\partial (x^2)}{\partial x}+\dfrac{\partial (-y)}{\partial y}+\dfrac{\partial (z)}{\partial z}=2x-1+1=2x$
The Divergence Theorem states that $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $
$=\iint_{y^2+z^2 \leq 9} \int_0^2 2xdxdydz$
$=\iint_{y^2+z^2 \leq 9} [x^2]_0^2 dydz$
$=\iint_{y^2+z^2 \leq 9} (4) dydz$
$=(4) \times 9 \pi$
$=36 \pi$
