trigonometry - Polar coordinates and extremes of integration

I have trouble understanding how finding the extremes of integration of $\theta$ when I pass in polar coordinates.

1° example - Let $(X,Y)$ a random vector with density $f(x,y)=\frac{1}{2\pi}e^{-\frac{(x^2+y^2)}{2}}$.

Using the transformation $g=\left\{\begin{matrix} x=rcos\theta\\ y=rsin\theta \end{matrix}\right.$ and after calculating the determinant of Jacobian matrix, I have $dxdy=rdrd\theta$ from which

$\mathbb{E}[g(X^2+Y^2)]=\int_{\mathbb{R}^2}g(x^2+y^2)f(x,y)dxdy=\frac{1}{2\pi}\int_{0}^{+\infty}g(r^2)e^{-\frac{r^2}{2}}\int_{0}^{2\pi}d\theta$
$\Rightarrow X^2+Y^2\sim Exp(\frac{1}{2})$

2° example - Why for $\int\int_{B}\frac{y}{x^2+y^2}dxdy$ with $B$ annulus of centre $(0,0)$ and radius $1$ and $2$ the extremes of integration of $\theta$ are $(0,\pi)$?

3° example - Why for $\int\int_{B}\sqrt{{x^2+y^2}}dxdy$ with $B$ segment of circle $(0,0)$ and radius $1$ and $2$ the extremes of integration of $\theta$ are $(0,\frac{\pi}{2})$?

4° example - Why for $\int\int_{S}(x-y)dxdy$ with $S={((x,y)\in \mathbb{R}:x^2+y^2=r^2; y\geq 0)}$ the extremes of integration of $\theta$ are $(0,\pi)$?

I hope I have made clear my difficulties.
Thanks in advance for any answer!