Generating the Volume of an n-Sphere

Date: 02/07/2002 at 03:29:54
From: Elara
Subject: Multivariable calculus

Hi,

Could you help me in finding an equation for the integration of a 
unit-ball in n-dimensions?

Date: 02/07/2002 at 10:15:41
From: Doctor Mitteldorf
Subject: Re: Multivariable calculus

If you know the volume of an n-1-sphere, you can generate the volume 
of an n-sphere with an integral. 

Let's set the radius = 1. Imagine stacking up the (n-1)-spheres, each 
of which has a radius (1-r^2), and integrating from -1 to 1. For 
example, if pi*r^2 is the answer for n=2, then for n=3 we have

  [Integral from -1 to +1] pi*(1-r^2) dr

  This gives the familiar answer V[3]=4pi/3, given that V[2]=pi.

For the next step, n=4, we have

  [Integral from -1 to +1] 4pi/3 * (1-r^2)^(3/2) dr

  This gives (1/2)pi^2 for the 4-sphere.

  In general, you can write

    V[n+1] = V[n] * [Integral from -1 to +1] (1-r^2)^(n/2) dr

If you want a general (non-recursive) formula, it's tricky because 
each of these integrals is a little different. The answer is worked 
out in terms of the gamma function in Kevin Brown's MathPages:

   Volume of n-Spheres and the Gamma Function
   http://www.mathpages.com/home/kmath163.htm   

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/08/2002 at 09:18:26
From: Elara
Subject: Multivariable calculus

Thank you for your help.  

Could you tell me if we can equate the volume of a unit n-ball in 
(n-1)dimensions with the area of the same ball in n-dimensions?

Yours,
Elara

Date: 02/08/2002 at 10:13:56
From: Doctor Mitteldorf
Subject: Re: Multivariable calculus

The n-dimensional volume of an ball of radius r is related to the 
n-1-dimensional surface area by the following formula:

     volume = area/(nr)

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/