Question

Find the volume of the ellipsoid x 2 a 2 + y 2 b 2 + z 2 c 2 = 1

Answer 1

Use a modified version of spherical coordinates.


`x(\rho,\theta,\varphi)=a\rho\cos\theta\sin\varphi`

`y(\rho,\theta,\varphi)=b\rho\sin\theta\sin\varphi`

`z(\rho,\theta,\varphi)=c\rho\cos\varphi`

The Jacobian for this change of coordinates is


`\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}`

for which
`|\det\mathbf J|=abc\rho^2\sin\varphi`.

Denoting the space bounded by the ellipsoid by
`\mathcal V`, the volume is given by the volume integral


`\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`