Answer
$\dfrac{384 \pi}{5}$
Work Step by Step
The Divergence Theorem states that $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $
$div F=\dfrac{\partial a}{\partial x}+\dfrac{\partial b}{\partial y}+\dfrac{\partial c}{\partial z}$
This implies that $div F=\dfrac{\partial (x^3+y^3)}{\partial x}+\dfrac{\partial (y^3+z^3)}{\partial y}+\dfrac{\partial (z^3+x^3)}{\partial z}=3x^2+3y^2+3z^2$
We get the spherical coordinates: $dV=dxdydz=\rho^2 \sin \phi d \rho d\theta d\phi$
$\iiint_E (3x^2+3y^2+3z^2)dV=\int_{0}^{\pi}\int_0^{2 \pi} \int_0^{2} (3 \rho^2) \rho^2 \sin \phi d\rho d\theta d\phi$
$=\int_{0}^{\pi}\int_0^{2 \pi} \int_0^{2} (3 \rho^4) \sin \phi d\rho d\theta d\phi$
$=3 [-\cos \phi]_{0}^{\pi} [\theta]_0^{2 \pi} [\dfrac{\rho^5}{5}]_0^2$
$=(3)\times (2) \times (2 \pi) \times \dfrac{32}{5}$
$=\dfrac{384 \pi}{5}$
