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

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

which has determinant

$\det\mathbf J=-\dfrac3{512}\rho^2\sin\varphi$

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

$\displaystyle\frac3{512}\int_{\varphi=0}^{\varphi=\pi}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=0}^{\rho=1}\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi=\dfrac\pi{128}$