Volume of a Hypersphere

Date: 06/03/99 at 17:47:20
From: Krishna Vedula
Subject: Volume of a hypersphere

Could you give me the formula for the volume of a hyper-sphere?

Thanks in advance
Krishna Vedula

Date: 06/04/99 at 05:24:40
From: Doctor Mitteldorf
Subject: Re: Volume of a hypersphere

Dear Krishna,

You can get the formula for a 3-d sphere by integrating the surface 
Area of a spherical shell from 0 out to R: Integral(4pi r^2 dr) = 
4/3 pi r^3.

What is the 4-dimensional volume of a "hypersphere"? You must first 
find the surface of a 3-d "shell," consisting of all points satisfying 
(w^2+x^2+y^2+z^2) = r^2.

How to get the area of the shell? Go back to how we got the area of a 
sphere in the first place. Imagine circles of latitude around the 
North Pole. Integrate these circles Southward along the earth's 
surface. The radius of each circle is R sin(theta). You can take 
circles of circumference 2pi R sin(theta) and integrate along the 
direction of theta from 0 to pi to get the area of the sphere, 
4 pi R^2.

Similarly, you can take spherical shells of radius R sin(theta) and 
area 4 pi R^2 sin^2(theta) and integrate from theta = 0 to pi halfway 
around the circumference of a hypersphere. This integral is 
2 pi^2 r^3.

Now we have the hyper-area of the hypersphere of radius r, and we can 
integrate from r = 0 to r = R, to get 1/2 pi^2 R^4 as the volume of 
the 4-dimensional hypersphere.

The volume of a sphere is a little more than half the volume of the 
circumscribed cube. The volume of a 4-d hypersphere is less than 1/3 
of the volume of the circumscribed hypercube.

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