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[parent] volume of the $n$-sphere (Derivation)

The volume contained inside $ S^n$, the $ n$-sphere (or hypersphere), is given by the integral

$\displaystyle V(n) = \int_{\sum_{i=1}^{n+1}x_i^2\le1} d^{n+1} x. $
Going to polar coordinates ( $ r^2=\sum_{i=1}^{n+1}x_i^2$) this becomes
$\displaystyle V(n) = \int_{S^n} d\Omega \int_0^1 r^{n}\, dr. $
The first integral is the integral over all solid angles subtended by the sphere and is equal to its area $ A(n)=\frac{2\pi^{\frac{n+1}{2}}}{\Gamma\left(\frac{n+1}{2}\right)}$, where $ \Gamma(x)$ is the gamma function. The second integral is elementary and evaluates to $ \int_0^1 r^{n}\, dr = 1/(n+1)$.

Finally, the volume is

$\displaystyle V(n) = \frac{\pi^{\frac{n+1}{2}}}{\frac{n+1}{2}\Gamma\left(\frac{n+1}{2}\right)} = \frac{\pi^{\frac{n+1}{2}}}{\Gamma\left(\frac{n+3}{2}\right)}. $
If the sphere has radius $ R$ instead of $ 1$, then the correct volume is $ V(n)R^{n+1}$.

Note that this formula works for $ n\ge0$. The first few cases are

$ n=0$
$ \Gamma(3/2)=\sqrt{\pi}/2$, hence $ V(0)=2$ (this is the length of the interval $ [-1,1]$ in $ \mathbb{R}$);
$ n=1$
$ \Gamma(2)=1$, hence $ V(1) = \pi$ (this is the familiar result for the area of the unit circle);
$ n=2$
$ \Gamma(5/2)=3\sqrt{\pi}/4$, hence $ V(2) = 4\pi/3$ (this is the familiar result for the volume of the unit sphere);
$ n=3$
$ \Gamma(3)=2$, hence $ V(3) = \pi^2/2$.



"volume of the $n$-sphere" is owned by CWoo. [ owner history (1) ]
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See Also: area of the $n$-sphere


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Cross-references: unit circle, interval, length, radius, gamma function, area, sphere, solid angles, polar coordinates, integral, hypersphere, contained, volume

This is version 7 of volume of the $n$-sphere, born on 2003-07-23, modified 2006-10-18.
Object id is 4496, canonical name is VolumeOfTheNSphere.
Accessed 9234 times total.

Classification:
AMS MSC: 51M05 (Geometry :: Real and complex geometry :: Euclidean geometries and generalizations)
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