Use a modified version of spherical coordinates.
[tex]x(\rho,\theta,\varphi)=a\rho\cos\theta\sin\varphi[/tex]
[tex]y(\rho,\theta,\varphi)=b\rho\sin\theta\sin\varphi[/tex]
[tex]z(\rho,\theta,\varphi)=c\rho\cos\varphi[/tex]
The Jacobian for this change of coordinates is
[tex]\mathbf J=\begin{bmatrix}a\cos\theta\sin\varphi&-a\rho\sin\theta\sin\varphi&a\rho\cos\theta\cos\varphi\\b\sin\theta\sin\varphi&b\rho\cos\theta\sin\varphi&b\rho\sin\theta\cos\varphi\\c\cos\varphi&0&-c\rho\sin\varphi\end{bmatrix}[/tex]
for which [tex]|\det\mathbf J|=abc\rho^2\sin\varphi[/tex].
Denoting the space bounded by the ellipsoid by [tex]\mathcal V[/tex], the volume is given by the volume integral
[tex]\displaystyle\iiint_{\mathcal V}\mathrm dV=abc\int_{\varphi=0}^{\varphi=\pi}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=0}^{\rho=1}\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\frac{4abc\pi}3[/tex]
