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.