Volume of a general ellipsoid

Question

Can somebody please show me at least one derivation of the volume of a general ellipsoid? I've been trying to derive by considering it a surface of revolution. The answer I keep getting is $4\pi(abc^3)/3$ but I know it's supposed $4\pi(abc)/3$. Could somebody show me at least one step by step derivation.

Answer 1

First, without loss of generality we can align the ellipsoid with the Cartesian axes.

Next, a unit sphere $S$ has volume $$\int_{S} 1\,dV = \frac{4}{3}\pi.$$

The ellipsoid is then given by the image of $S$ under the diffeomorphism $\phi(x,y,z) = (ax, by, cz)$. The volume of the ellipsoid is then

$$\int_{\phi(S)} 1\, dV = \int_S 1 \det \left[\begin{array}{ccc}a & 0 & 0\\0 & b & 0\\0 & 0 &c\end{array}\right]\, dV = abc\int_S 1\, dV = \frac{4abc\pi}{3}.$$

Answer 2

volume of $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ using integrals

$$2\int_{\theta=0}^{2\pi}\int_{r=0}^1\int_{z=0}^{c\sqrt{1-r^2}}|J|dz~dr~d\theta$$
where |J|=abr