Converting rectangular to polar coordinates

Question

Why when converting rectangular to polar is theta only $0$ to $\pi/2$ and not $0$ to $2\pi$?? problem I'm working with

Answer 1

The region of integration is a quarter disk of radius $2$ centered at the origin, in the first quadrant. In polar coordinates, this means integrating $r$ from $0$ to $2$, and $\theta$ from $0$ (the x-axis) through the first quadrant to $\pi/2$ (the y-axis).

Answer 2

You already know that the integration limits involve the disk centered on origin and radius $2$. If we look at the $y$ integral, we see that it goes from $0$ to $2$. That means that we deal with only the upper half of the disk, so we must restrict $\theta$ from $0$ to $\pi$. Now we look at $x$. Similarly, you only integrate from $0$, so no negative $x$, which means no $\theta>\pi/2$.

To get the integral from $0$ to $\pi$, the $x$ integration must go between $-\sqrt{4-y^2}$ to $\sqrt{4-y^2}$, and to extend $\theta$ to $2\pi$ you would need to change the $y$ integral limits to be $-2$ to $2$.