volume of ellipsoid

Let us determine the volume of the ellipsoid

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=\mathrm{\hspace{0.33em}1}.$$

Suppose  $-a\leqq x\leqq a$. When we cut the ellipsoid with a plane parallel to the $yz$-plane, that is, let $x$ be , we get the ellipse

$$\frac{y^2}{b^2}+\frac{z^2}{c^2}=\mathrm{\hspace{0.33em}1}-\frac{x^2}{a^2},$$

i.e.

$$\frac{y^2}{b^2\left(1-\frac{x^2}{a^2}\right)}+\frac{z^2}{c^2\left(1-\frac{x^2}{a^2}\right)}=\mathrm{\hspace{0.33em}1},$$

with the semiaxes

$$b_1:=b\sqrt{1-\frac{x^2}{a^2}},c_1:=c\sqrt{1-\frac{x^2}{a^2}}.$$

The area of this ellipse is $\pi b_1{c}_1$ (see area of plane region), and thus we have the function

$$A(x):=\pi bc\left(1-\frac{x^2}{a^2}\right)$$

expressing the area cut of the ellipsoid by parallel planes. By the volume formula of the parent entry (http://planetmath.org/VolumeAsIntegral) we can calculate the volume of the ellipsoid as

$$V={\int }_{-a}^{a}A(x)𝑑x=\pi bc{\int }_{-a}^a\left(1-\frac{x^2}{a^2}\right)𝑑x=\pi bc\underset{x=-a}{\overset{a}{/}}\left(x-\frac{x^3}{3a^2}\right)=\frac{4}{3}\pi abc.$$

The special case  $a=b=c=r$  of a sphere is the well-known expression $\frac{4}{3}\pi r^3.$

Title	volume of ellipsoid
Canonical name	VolumeOfEllipsoid
Date of creation	2013-03-22 17:20:41
Last modified on	2013-03-22 17:20:41
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	11
Author	pahio (2872)
Entry type	Result
Classification	msc 51M25
Synonym	ellipsoid volume
Related topic	Ellipsoid
Related topic	SubstitutionNotation
Related topic	SqueezingMathbbRn