Question

Find the volume of the ellipsoid x^2+y^2+9z^2=64

Answer 1

Parameterize the ellipsoid using the augmented spherical coordinates:


`\begin{cases}x=\frac18\rho\cos\theta\sin\varphi\\\\y=\frac18\rho\sin\theta\sin\varphi\\\\z=\frac38\rho\cos\varphi\end{cases}`

Then the Jacobian for the change of coordinates is


`\mathbf J=(\partial(x,y,z))/(\partial(\rho,\theta,\varphi))=\begin{bmatrix}\frac18\cos\theta\sin\varphi&-\frac18\rho\sin\theta\sin\varphi&\frac18\rho\cos\theta\cos\varphi\\\\\frac18\sin\theta\sin\varphi&\frac18\rho\cos\theta\sin\varphi&\frac18\rho\sin\theta\cos\varphi\\\frac38\cos\varphi&0&-\frac38\rho\sin\varphi\end{bmatrix}`

which has determinant


`\det\mathbf J=-\frac3{512}\rho^2\sin\varphi`

Then the volume of the ellipsoid is given by


`\displaystyle\iiint_E\mathrm dx\,\mathrm dy\,\mathrm dz=\iiint_E|\det\mathbf J|\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi`

where
`E` denotes the spaced contained by the ellipsoid. In particular, we have the definite integral and volume


`\displaystyle\frac3{512}\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\pi{128}`