circular corral of unit radius is enclosed by a fence. A goat is outside the corral and tied to the fence with a rope of length $a \le \pi$ (see the figure to the right). What is the area of the region outside the corral that the goat can reach?\\
Let the point at which the goat is tied to the fence be the origin, and let the positive x-axis point outward from the corral. There are three regions to consider, one that is to the right of the vertical line $\theta = \frac{\pi}{2}$ and one in each of quadrants II and III. The area of the region to the right of the vertical line $\theta = \frac{\pi}{2}$ is $\frac{a^2}{2}$.\\(Type an exact answer.)\\The areas of the regions in quadrants II and III are equal. Consider only the region in quadrant II. Assume the goat walks around the corral while keeping the rope taut and let $\phi$ be the angle with vertex at the center of the corral, one side through the origin, and one side through the farthest point on the fence that the goat's rope is touching as he moves. For a given value of $\phi$, the length of rope along the fence is $\phi$ and the length of the rope not along the fence is $a - \phi$.\\Set up the integral that gives the area of the region in quadrant II. Select the correct choice below and fill in the answer box to complete your choice.\\A. $\int_0^a$ $d\phi$\\B. $\int_0^{\pi}$ $d\phi$\\C. $\int_0^1$ $d\phi$
