Volume of n-dimensional sphere

[Superposed_Cat]

Hello, this may seem like a stupid question but how would one calculate the volume of an n-dimensional sphere?
Thanks.

 

[jedishrfu]

I think you would extroplate the formulas for area of a circle to volume of a sphere to hypervolume of a hypersphere...

pi*r^2
4/3pi*r^3

http://en.wikipedia.org/wiki/N-sphere

Midway through the article is a cool table of the progress of n from 0 to ... and the volumes and surfaces of the hyperspheres.

 

[JJacquelin]

Hi,

One can find the formulas of volume of the hyperspheres, hypercones, hypersectors, in : http://fr.scribd.com/doc/14789360/Le-probleme-de-lhyperchevre-

 

[HallsofIvy]

Given an x_1,x_2, ..., x_n Cartesian coordinate system, the the equation of the n-sphere of radius R, with center at the origin is x_1^2+ x_2^2+ x_3^2+ \cdot\cdot\cdot+ x_n^2= R^2.
It is clear that, if all the other variables are 0, then x_1^2= R^2 so that x_1 ranges between -R and R to cover the entire n-sphere. In the x_1x_2 plane, all other variables 0, x_1^2+ x_2^2= R^2 so that, for fixed x_1, x_2= \pm\sqrt{R^2- x_1^2} and so x_2 ranges between -\sqrt{R^2- x_1^2} and \sqrt{R^2- x_1^2} etc.

Continuing like that, we see that the volume is given by
\int_{-R}^R\int_{-\sqrt{R^2- x_1^2}}^{\sqrt{R^2- x_1^2}}\int_{-\sqrt{R^2- x_1^2- x_2^2}}^{\sqrt{R^2- x_1^2- x_2^2}}\cdot\cdot\cdot\int_{-\sqrt{R^2- x_1^2- x_2^2- \cdot\cdot\cdot- x_{n-1}^2}}^{\sqrt{R^2- x_1^2- x_2^2- \cdot\cdot\cdot- x_{n-1}^2}} dx_ndx_{n-1}\cdot\cdot\cdot dx_2 dx_1.

You ought to be able to take the formulas for area of a circle (2-sphere), volume of a sphere (3-sphere) and use that integral to find the hyper-volumes of the 4-sphere, 5-sphere, etc to find a general formula.

 

[mathman]

http://en.wikipedia.org/wiki/N-sphere

Above is a detailed discussion. Note that the development works with a generalization of spherical coordinates.

 

[Superposed_Cat]

Thanks all especially halls of ivy as I was just about to ask for it in integral form