= Solution
For a bounded region $V$ with piecewise smooth closed boundary $S$, outward unit <normal vector> $\boldsymbol n$ and continuously differentiable <vector field> $\boldsymbol F$, the <divergence theorem> states $\int_S\boldsymbol F\cdot\boldsymbol n\,dS=\int_V\nabla\cdot\boldsymbol F\,dV$. Here $dS$ is scalar <surface area>, $dV$ is <volume>, and $d\boldsymbol S=\boldsymbol n\,dS$ is the oriented <area element>.
For $\boldsymbol F=\boldsymbol r$, its <divergence> is three. The <ellipsoid> has semiaxes $a,a,b$, so its <volume> is $4\pi a^2b/3$, obtained by scaling the unit ball. Consequently
$$
\boxed{\int_S\boldsymbol r\cdot d\boldsymbol S=3\operatorname{Vol}(V)=4\pi a^2b.}
$$
Back to article page
