Let $\mathbf{F}(x, y, z) = \langle P, Q, R \rangle$ be a smooth vector field defined on $\mathbb{R}^3$. Suppose $\nabla \cdot \mathbf{F} = x^2 + y^2 + z^2$ and the flux through the surface of a sphere $S$ of radius $a$ centered at the origin is $\Phi$. Which of the following expressions correctly represents $\Phi$ using the Divergence Theorem?