How can I calculate the volume of an elipsoid using triple integrals?

[PsychonautQQ]

Homework Statement

Find the volume of Elipsoid x^2+y^2+5z^2=16

The Attempt at a Solution

So if x and y are both zero z goes from -(16/5)^(1/2) to (16/5)^(1/2)
and if I do it in polar coordinates then r goes from 0 to 4
and theta goes from 0 to 2pi?

so Triple Integral: rdrdθdz with the above parameters should give me the correct answer... right?

 

[vanhees71]

How about parametrizing the ellipsoid as
$$\vec{r}(\lambda,\theta,\phi)=\lambda \begin{pmatrix}<br /> a \cos \phi \sin \theta \\<br /> b \sin \phi \sin \theta \\<br /> c \cos \theta<br /> \end{pmatrix}?$$
Now you have to find the boundaries of the three parameters and check that it's really giving the ellipsoide. Then evaluate the Jacobian and do the integral :-).

 

[PsychonautQQ]

ah is there a way to do it with cylindrical coordinates like the way I was doing it? I think the problem wants me to use this method X_x.

 

[haruspex]

ah is there a way to do it with cylindrical coordinates like the way I was doing it? I think the problem wants me to use this method X_x.

Yes, but z and r are not independent. The value of one will affect the bound on the other.
(Far the easiest way here is to substitute t = z*constant, if you pick the right constant.)