# Integral homework problem help

• -=nobody=-

#### -=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?

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

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

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

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!

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

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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?

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

$$x= a r \cos\theta$$
$$y = b r\sin \theta$$
Now, find the Jacobian and limits of integration of $\theta$ and $r$. Can you take it from here?