volume of ellipsoid

Let us determine the volume of the ellipsoid

 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}\;=\;1.$

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

 $\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}\;=\;1\!-\!\frac{x^{2}}{a^{2}},$

i.e.

 $\frac{y^{2}}{b^{2}\left(1\!-\!\frac{x^{2}}{a^{2}}\right)}+\frac{z^{2}}{c^{2}% \left(1\!-\!\frac{x^{2}}{a^{2}}\right)}\;=\;1,$

with the semiaxes

 $b_{1}:=b\sqrt{1\!-\!\frac{x^{2}}{a^{2}}},\quad c_{1}\;:=\;c\sqrt{1\!-\!\frac{x% ^{2}}{a^{2}}}.$

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

 $V\;=\;\int_{-a}^{a}\!A(x)\,dx=\pi bc\int_{-a}^{a}\!\left(1\!-\!\frac{x^{2}}{a^% {2}}\right)\,dx\;=\;\pi bc\!\operatornamewithlimits{\Big{/}}_{\!\!\!x\,=-a}^{% \,\quad a}\left(x-\frac{x^{3}}{3a^{2}}\right)\;=\;\frac{4}{3}\pi abc.$

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 VolumeOfEllipsoid 2013-03-22 17:20:41 2013-03-22 17:20:41 pahio (2872) pahio (2872) 11 pahio (2872) Result msc 51M25 ellipsoid volume Ellipsoid SubstitutionNotation SqueezingMathbbRn