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 $\frac{4}{3}\pi r^3.$