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- Jan 26, 2012

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**Background Info**: 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 seen in Figure 1, then the parametric equations for the involute of the circle are

\[\left\{\begin{aligned} x(\theta) &= r \left(\cos\theta +\theta\sin\theta\right) \\ y(\theta) &= r\left(\sin\theta - \theta\cos\theta\right)\end{aligned}\right.\]

**Problem**: A cow is tied to a silo with radius $r$ by a rope just long enough to reach the opposite side of the silo. Find the area available for grazing by the cow.

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