A string is wound around a circle and then unwound while being held taut. The curve traced by the point $P$ at the end of the string is called the involute of the circle. If the circle has radius $r$ and center $O$ and the initial position of $P$ is $(r, 0)$, and if the parameter $\theta$ is chosen as in the figure below, then it can be shown that the parametric equations of the involute are $$x=r(\cos(\theta)+\theta \sin(\theta)),\quad y=r(\sin(\theta)-\theta \cos(\theta)).$$ (You should check this to make sure you understand how to derive these parametric equations!)

A cow is tied to a silo with radius $r$ by a rope just long enough to reach the opposite side of the silo (see the figure below). Find the area available for grazing by the cow.

Grazing area =