Let [tex]R[/tex] be the ellipsoid with equation
[tex]\left(\dfrac xa\right)^2+\left(\dfrac yb\right)^2+\left(\dfrac zc\right)^2=1[/tex]
so that the volume of [tex]R[/tex] is given by the triple integral
[tex]\displaystyle\iiint_R\mathrm dV[/tex]
Consider the augmented spherical coordinates given by the identities
[tex]\begin{cases}x=ar\cos u\sin v\\y=br\sin u\sin v\\z=cr\cos v\end{cases}[/tex]
Computing the Jacobian, we find that the volume element is given by
[tex]\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=abcr^2\sin v\,\mathrm dr\,\mathrm du\,\mathrm dv[/tex]
so that the volume integral can be written as
[tex]\displaystyle\iiint_R\mathrm dV=abc\int_{v=0}^{v=\pi}\int_{u=0}^{u=2\pi}\int_{r=0}^{r=1}r^2\sin v\,\mathrm dr\,\mathrm du\,\mathrm dv=\frac{4abc\pi}3[/tex]
