1. ## double integral

any help would much be appreciated!

Find the area of the region enclosed by the ellipse $x^2/a^2 + y^2/b^2 =1$

2. Let $R=\left\{ (x,y)\in \mathbb{R}^{2}\bigg|\frac{x^{2}}{a^{2}}+\frac{y^{2 }}{b^{2}}\le 1,\,a,b>0 \right\}.$

Put $(u,v)=\left( \frac{x}{a},\frac{y}{b} \right),$ whereat $du\,dv=\frac{dx\,dy}{ab}$ and the region $R$ will be transfered to the unit circle $u^{2}+v^{2}\le 1.$

Switch to polar coordinates and we're done.

3. I don't understand could you please explain more.

4. Unless you don't tell what's exactly you don't understand, I can't help further.

5. ok I do get it now, but then i'm stuck i get
and the Jacobian is and so then

6. You can swich to polar to integrate this.
BUT as the previous poster said, it's now a circle with radius one.
Thus the area inside of it is PI.
Hence, there's no need to integrate.
Giving you $ab\pi$ as the answer.
This works in all dimensions.

7. switching to polar what would we get

8. Originally Posted by sonia1
switching to polar what would we get

There is no reason to integrate.
The area of a circle with radius one is $\pi$.

9. but we've been asked to use double integration