Need surface area of N-Ellipsoid.

Hi guys.
What is surface area of N dimensional ellipsoid?
Any help is really appreciated.

 

Re: Welcome to PF!

I tried. But I'm getting something which is not easily integrable

 

Re: URGENT! Need surface area of N-Ellipsoid.

Suppose that ellipsoid has axis a_{1},...,a_{N}.
Then S=[tex]\int_{{\sum\frac{x^{2}_{i}}{a^{2}_{i}}<1}[/tex]dS(x)-surface integral.
Then i solved for x_{N}=a_{N}([tex]\sqrt{1-(\frac{x_{1}}{a_{1}})^{2}-...-(\frac{x_{N-1}}{a_{N-1}})^{2}}[/tex]) and used that formula for evaluating the surface area, where you need evaluate N-1 dimensional volume integra (Found the partial derivatives of x_{N} w/r to x_{i} and etc). IN that integral i made a substitution , i.e. linear transformation x_{i}=y_{i}a_{i}, i=1,...N-1. I got:

Integral over {B(0,1)} of[tex] {\sqrt{1+((\frac{a_{N}}{a_{1}})^{2}-1)y^{2}_{1}+...+((\frac{a_{N}}{a_{N-1}})^{2}-1)y^{2}_{N-1}}} [/tex]*Jac(Transformation), where B(0,1) is n-1 unit ball.
I think i made a mistake, but i can't find it. Thanks and by the way, what is the good textbook to brush up on these things

 

Re: URGENT! Need surface area of N-Ellipsoid.

Actually, I used same idea to find the volume and I got it:
it is a_{1}*...a_{N}*meas(unit ball)

 

That doesn't make a sense. Because, if a_{i}=r for all i's, we don't get the surface area ball with radiuis r.

 

OK. Where is my mistake? Thanks
i'm having trouble with finding it

 

I see what you are saying, but at least did I set it up correctly?

 

Could you please tell me how you got it?