How Do You Solve This Elliptical Integral with a Coordinate Transformation?

[-=nobody=-]

$$2c\int_{x=-a}^a\int_{y=-b\sqrt{1-\frac{x^2}{a^2}}}^{b\sqrt{1-\frac{x^2}{a^2}}}\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}dydx$$
Can you help me with this integral?

 

[arildno]

You have already been advised to do a change of variables, rather than do this in Cartesian variables.

 

[-=nobody=-]

Should I do it like it is described in this article?
http://libraryofmath.com/math/Calcu...mple_Calculus_III_Volume_of_an_Ellipsoid.html
If I do it so u=x/a, v=y/b, w=z/c.
How can I prove that http://libraryofmath.com/math/Calcu...f_Variables_in_Multiple_Integrals_gr_136.gif" =3/4Pi*a*b*c

 

[arildno]

Well, how would you go about proving that the unit ball has volume $\frac{4}{3}\pi$ ?

 

[-=nobody=-]

Well, the idea is probably good, but it doesn't help me with the integral

 

[HallsofIvy]

Well, the idea is just to use spherical coordinates. Have you sketched the region over which that integral is taken? Looks to me like there is a heckuvalot of symmetry there!

 

[-=nobody=-]

Won't it be much more complicated, or is it the only way?
r=(x^2+y^2+z^2)^1/2.

 

[-=nobody=-]

It's clear that there is symmtry, but if r=(x^2+y^2+z^2)^1/2 everything will be much more complicated. How should it be solved then?

 

[-=nobody=-]

Sorry for this post, I had some problems with my internet browser.

 

[siddharth]

-=nobody=-, as everyone has already said on this thread, transform your coordinates. ie, set

$$x= a r \cos\theta$$

$$y = b r\sin \theta$$

Now, find the Jacobian and limits of integration of $\theta$ and $r$. Can you take it from here?