Answer
$\dfrac{4\pi}{3}$
Work Step by Step
Divergence Theorem: $\iiint_Ediv \overrightarrow{F}dV=\iint_S \overrightarrow{F}\cdot d\overrightarrow{S} $
where, $div F=\dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z}=\dfrac{\partial (2)}{\partial x}+\dfrac{\partial (2)}{\partial y}+\dfrac{\partial z}{\partial z}=1$
In the Divergence Theorem the integral $\iiint_E div F dV$ defines the volume of the region $E$ and $E$ lies inside a sphere with radius $1$.
Volume of the region E: $\iiint_E dV=\dfrac{4\pi(1)^3}{3}=\dfrac{4\pi}{3}$
This implies that $\iint_S F \cdot n dS=\iiint_Ediv \overrightarrow{F}dV =\dfrac{4\pi}{3}$
