Integrating over sphere

I am trying to solve integral

$$I=\int \frac{dS}{\sqrt{\frac{x^2}{a^2}+\frac{y^2}{a^2}+\frac{z^2}{b^2}}}$$ over a sphere, where $$r = \sqrt{\frac{x^2}{a}+\frac{y^2}{a}+\frac{z^2}{b}}$$.

I thought about using generalized spherical coordinates like $$\frac{x}{\sqrt{a}}=r \cos{\phi}\sin{\theta}$$ etc. but $$a$$ and $$b$$ are squared in the integral...

How to solve this integral?

• by sphere here you mean the ball (the volume of the sphere) or just the surface? – Masacroso Apr 21 at 12:58
• just the surface – Andrej Apr 21 at 16:10

1 Answer

Ok, sorry about the last comment. I tried crossing things out, it didn't work.

I think that the best that can be done is the following substitution:

$$\left\{ \begin{array}{c} x=arcos(\theta) \\ y=arsin(\theta) \\ z=bv \end{array} \right.$$ And then integrating with respect to r first then z then $$\theta$$

Because the determinant of the determinant of the Jacobian has an r in it.

And then you need to separate the two the integrals, and do a trigonometric substitution to each one alone.

Final note: the initial integrand is independent of $$\theta$$ so you can multiply by 2$$\pi$$ from the beginning.