Volume of a HypersphereDate: 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/ |
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