Use polar coordinates to describe the region $R$ representing the quarter circle in the first quadrant of the $xy$-plane
The solution to this problem is that $$R = \left\{(r, \theta): 0 \le r\le 2, \, 0\le \theta \le \frac{\pi}{2}\right\}$$
I don't have the ability to post a picture, but note that I am talking about a shaded region, used for finding the area in multivariate calculus.
Here's my question: why does $r$ span from $0$ to $2$? In a quarter circle with radius $2$, the radius is always equal to $2$, is it not? So why exactly is the region claiming $0\le r \le2$? Here's a more concrete example:
The area of a quarter circle in the first quadrant is given by: $$\int_0^{\pi/2}\int_0^2r\,drd\theta.$$
Why do we integrate $r$ from $0$ to $2$? Pretty obvious that due to the definition of integration, going from $2$ to $2$ would yield a zero area, but what is the logic behind $0$ to $2$? Thanks.