Given a region where the $x$ limits are $-1< x<1$ and $0< y<\sqrt{4-x^2}$, with the option of converting into polar coordinates, i.e. the function $(x,y)$ can be replaced by $r^2$.
I'm finding it tricky to get the limits for $r$. The region $R$ would look like a semi-circle bounded below by the $x$-axis, with radius $2$. But the $x$ limits are $1$ and $-1$, so the region we want is inside of this semi-circle?
Is it possible to approximate this as a rectangle and take the $r$-limits as $-1,1$ and $\theta$ as $0,\pi$?
Any help would be much appreciated
$R$ can be seen as the disjoint union of three sets, $R=R_1\cup R_2\cup R_3$, where \begin{align*} R_1&=\{(r,\theta)\in\mathbb{R}^2:0<\theta<\pi/3,0<r<\sec\theta\}\\ R_2&=\{(r,\theta)\in\mathbb{R}^2:\pi/3\le\theta<2\pi/3,0<r<2\}\\ R_3&=\{(r,\theta)\in\mathbb{R}^2:2\pi/3\le\theta<\pi,0< r <|\sec \theta|\} \end{align*}