Use the divergence theorem to calculate the surface integral $\iint_S \mathbf{F} \cdot d\mathbf{S}$; that is, calculate the flux of $\mathbf{F}$ across $S$.
$\mathbf{F} = |\mathbf{r}|\mathbf{r}$, where $\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$, $S$ consists of the hemisphere $z = \sqrt{9 - x^2 - y^2}$ and the disk $x^2 + y^2 \le 9$ in the $xy$-plane