# Tough integration with change of variables and switch to polar coordinates

I was given this question in class and I was just wondering if I am on the right track… Evaluate: $$I=\iint\left(1-\frac{x^2}{a^2} -\frac{y^2}{b^2} \right)^{3/2} dxdy$$ over the region enclosed by ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} =1 .$$

First I changed $x=au$ and $y=bv$. Therefore boundary -> $u^2+v^2 \leq1$.

Also I changes $I$ to

$$I=\iint(1-u^2-v^2)^{3/2} ab \;dudv$$ From here I switched to polar coordinates and solved the integral. $$ab\int_0^{2\pi}d\phi \int_0^1 r(1-r^2)^{3/2} dr$$

After solving this I ended up with $\frac{2ab}{5}\pi$… Does this look correct? I was just confused about changing variables then applying polar coordinates and was wondering if I made a mistake there. Any help is greatly appreciated.

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It is correct. The problem was done precisely as it should be done. –  André Nicolas Mar 25 '13 at 18:11
Kudos to Andre that was able to understand that mess. @user68203, go to the FAQ section to find directions on how to write LaTeX in this site to properly write mathematics in this site. –  DonAntonio Mar 25 '13 at 18:26
You may answer your own question. In addition to DonAntonio's comment you find some basic information about writing math at this site here, here, here and here. –  Américo Tavares Mar 25 '13 at 18:58
I've changed ∅ to $\phi$ (\phi) in the last integral. However as you likely know in polar coordinates the standard variable is $\theta$ (\theta). –  Américo Tavares Mar 25 '13 at 19:15
Thank you guys. And yes sorry for the mess. I will check out those sites for next time. –  user68203 Mar 26 '13 at 0:42