any help would much be appreciated!
Find the area of the region enclosed by the ellipse $\displaystyle x^2/a^2 + y^2/b^2 =1$
Let $\displaystyle R=\left\{ (x,y)\in \mathbb{R}^{2}\bigg|\frac{x^{2}}{a^{2}}+\frac{y^{2 }}{b^{2}}\le 1,\,a,b>0 \right\}.$
Put $\displaystyle (u,v)=\left( \frac{x}{a},\frac{y}{b} \right),$ whereat $\displaystyle du\,dv=\frac{dx\,dy}{ab}$ and the region $\displaystyle R$ will be transfered to the unit circle $\displaystyle u^{2}+v^{2}\le 1.$
Switch to polar coordinates and we're done.
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 $\displaystyle ab\pi$ as the answer.
This works in all dimensions.