Answer
$\dfrac{256\pi}{3}$
Work Step by Step
$div F=\dfrac{\partial a}{\partial x}+\dfrac{\partial b}{\partial y}+\dfrac{\partial c}{\partial z}$
Now, $div F=0+1+0=1$
The Divergence Theorem states that $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $
Here, we have $\iiint_E div F dV$ is the volume of the region $E$. In the given problem, $E$ lies inside a sphere having radius $4$.
Thus, the volume of the region E is equal to $=\dfrac{4\pi(4)^3}{3}=\dfrac{256\pi}{3}$
Hence, $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV =\dfrac{256\pi}{3}$
