# Calculate the integrals $\int_{\partial E}\rho \, dS$ and $\int_{\partial E} \frac{1}{\rho} dS$ over an ellipsoid

Let $E= \{\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} \le 1\}$ be a solid ellipsoid. For every point $p \in \partial E$ we define $\rho(p)$ to be the distance between the origin and the affine tangent space at $p$.

Calculate the integrals $\displaystyle \int_{\partial E}\rho \, dS$ and $\displaystyle \int_{\partial E} \frac{1}{\rho} \, dS$.

I was able to calculate the first, I got $3\operatorname{Vol}(E)$. However, I have no idea about the second. To do that I found the perpendicular normal vector $n(p)$ and then $\rho(p) = \langle p,n(p) \rangle$ and used the divergence theorem.

Generally, the solution should be related to those subjects (divergence, flux).

Any tip or advice would be appreciated.

Thanks!

Equation of tangent plane at $(x',y',z')$
$$\frac{x'x}{a^2}+\frac{y'y}{b^2}+\frac{z'z}{c^2}=1$$
$$\rho= \frac{1}{\sqrt{\dfrac{x'^2}{a^4}+\dfrac{y'^2}{b^4}+\dfrac{z'^2}{c^4}}}$$
Let $\begin{pmatrix} x' \\ y' \\ z' \end{pmatrix}= \begin{pmatrix} a\sin u \cos v \\ b\sin u \sin v \\ c\cos u \end{pmatrix}$