# Volume of Ellipsoid

Elsewhere we found the volume V of an ellipsoid \displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 with the help of the Cavalieri Principle. Hubert Shutrick has pointed out that using a special kind of coordinate makes common triple integration no more difficult.

\displaystyle V=\int\int\int dV

In elliptical polar coordinates

x = a r \mbox{cos}\theta \mbox{sin}\phi
y = b r \mbox{sin}\theta \mbox{sin}\phi
z = c r \mbox{cos}\phi

the element of volume is abcr^{2}\mbox{sin}\phi, so, when you integrate 0 \lt r \lt 1, 0 \lt \theta \lt 2\pi, 0 \lt \phi \lt \pi, you get abc\cdot \frac{1}{3}\cdot 2\pi\cdot 2, the usual formula, V=\frac{4\pi}{3}abc.