Answer
$\dfrac{384 \pi}{5}$
Work Step by Step
Divergence Theorem: $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $
$div F=\dfrac{\partial p}{\partial x}+\dfrac{\partial q}{\partial y}+\dfrac{\partial r}{\partial z}=\dfrac{\partial (x^3+y^3)}{\partial x}+\dfrac{\partial (y^3+z^3)}{\partial y}+\dfrac{\partial (z^3+x^3)}{\partial z}=3x^2+3y^2+3z^2$
Apply Spherical coordinates.
Since, $dV=dxdydz=\rho^2 \sin \phi d \rho d\theta d\phi$
Now, $I=\iiint_E (3x^2+3y^2+3z^2)dV=\int_{0}^{\pi}\int_0^{2 \pi} \int_0^{2} (3 \rho^2) \times \rho^2 \times \sin \phi d\rho d\theta d\phi$
and $I=\int_{0}^{\pi}\int_0^{2 \pi} \int_0^{2} (3 \rho^4) \times \sin \phi \times d\rho d\theta d\phi=3\times [-\cos \phi]_{0}^{\pi} \times [\theta]_0^{2 \pi} \times [\dfrac{\rho^5}{5}]_0^2$
Hence, $I=3\times [-(\cos \pi-\cos 0)] \times [2 \pi-0] \times [\dfrac{(2)^5}{5}-0]=(3)\times (1+1) \times (2 \pi) \times \dfrac{32}{5}=\dfrac{384 \pi}{5}$
