The $\theta r$ coordinate system and the $xy$ coordinate system are
related by $x=r\cos\theta$ and $y=r\sin\theta$. The small rectangle
in the $\theta r$ plane corresponds under the mapping to the
four-sided figure in the $xy$ plane.
The large rectangle in the $\theta r$ plane is $[0,\pi]\times[0,1]$ and
corresponds to the half-circle in the $xy$ plane.
Drag the
blue dot in the $\theta r$ plane to move the rectangle; drag the green
dot to resize the rectangle.