Calculating Volume of an Ellipsoid Using Spherical Coordinates

[PsychonautQQ]

Homework Statement

find the volume of the ellipsoid x^2+y^2+5z^2=16

The Attempt at a Solution

so I assume I must first change it to spherical coordinates. I figure I evaluate θ from 0 to 2∏ abd ∅ from 0 to ∏, but the p is giving me a challenge. if z is zero then p goes from 0 to 4, but if x and y are zero then p goes from 0 to (16/5)^(1/2)? am I on the right track? What do I do now??

 

[tiny-tim]

Hi PsychonautQQ!

so I assume I must first change it to spherical coordinates. …

(If you must integrate) wouldn't ordinary x,y,z integration be easier?

 

[vela]

Cylindrical coordinates would seem to be a better choice than spherical.

In either case, you want to express the outer boundary in terms of the coordinates you decide to work with. In the case for spherical coordinates, solve for $\rho$ as a function of $\phi$ to get the upper limit for $\rho$.

 

[LCKurtz]

@PsychonautQQ: Have you studied change of variables and Jacobians yet?

 

[vanhees71]

As with the analogous problem of the area of an ellipse, also here I'd use the most convenient parametrization possible. I'd suggest
$$\vec{r}(\lambda,\theta,\phi)=\lambda \begin{pmatrix} a \cos \phi \sin \theta \\ b \sin \phi \sin \theta \\ c \cos \theta \end{pmatrix}, \quad \lambda \in [0,1], \quad \phi \in [0, 2\pi], \quad \theta \in [0,\pi].$$
Calculate the Jacobian and do the triple integral. it's not too difficult.