Volume of N-Dimensional Ellipsoid and Its Extension

Volume of n-dimensional ellipsoid and its extension
Team Registration Number: 90014 School Name: Methodist College Teacher’s Name: Mr. Yeung Sik Ming Team Members’ Names: Wu Ming Hung, Tsang Man Ho, Lam Cheuk Yan, Mak Sze Long

A report submitted to the Scientific Committee of the Hang Lung Mathematics Award, 2010

7th August 2010

Abstract
In this paper, we want to find the formula of the volume of the n-dimensional ellipsoid. First, we find the volumes of the 2-dimensional and 3-dimensional ellipsoids by double and triple integrals. After that, we solve the multiple integral having integrand with different exponents e1 , e2 ,

 1  , en    ,   :  2 

   x  x 
En a 2 e1 1 2

2 e2

x  n 2 en

d n . dxn 

 whereEn a   x1, 



, xn  

n

:

 xk 2  1andd n  dx1dx2 2 k 1 ak  n Finally, from the result of the multiple integral, we can find the formula of the volume of the n-dimensional ellipsoid.

1

Introduction
For the first part of the project, we will find the volumes of 2-dimensional and 3-dimensional ellipsoids. This part is done in order to check for the consistency of the formula of the volume of n-dimensional ellipsoid. In the process of finding the volume of 3-dimensional ellipsoid, we will use the technique of multiple integration and the Jacobian matrix. After that, we are going to find the formula of the volume of n-dimensional ellipsoid. More generally, we consider the K function K x, en    xk 2  . Firstly, if the n ek k 1





exponents are zero, then the answer will be the volume of the n-dimensional ellipsoid. Secondly, if the integrand consists of linear combinations of K functions with different exponents, then by the linearity of integral, we can calculate the integral by separating it into a linear combination of integrals. Finally, we want to investigate the relation between K function and the probability density function of Dirichlet distribution.