Calculating the flux over the base of a hemiellipsoid

Question

I have this task where I'm supposed to calculate $\int_{D}((\textbf{B} \cdot \textbf{r})\textbf{C})d\textbf{S}$, where $D$ is hemiellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1, z>0$. I know that I can rewrite this with help of Gauss theorem, which gives me $$\int_{D}((\textbf{B} \cdot \textbf{r})\textbf{C})d\textbf{S} + \int_{D'}((\textbf{B} \cdot \textbf{r})\textbf{C})d\textbf{S} = \int_{V}\nabla \cdot((\textbf{B} \cdot \textbf{r})\textbf{C})d\textbf{V}.$$ The vectors $C$ and $B$ are constant vectors and $r$ is the position vector. $D'$ is the bottom plate of the hemiellipsoid when z=0. I've calculated this to be $2\pi abc/3+\int_{D'}((\textbf{B} \cdot \textbf{r})\textbf{C})d\textbf{S}$. But I don't seem to understand how I'm supposed to get the second integral with D' because of the vector I'm supposed to integrate.