Area of the field that the cow can graze. How do we find the area that the cow can graze?
The question goes as follows--
There is a circular barn house surrounded by a huge grazing field. A cow is tied to the rope ($AB$) at the end $A$ as shown. The length of the rope is half the circumference of the barn. Find the area that the cow can graze. 
The left side of the area is obvious but i cannot get a hang of what is happening on the right side..all i can say is that the rope starts to wrap around the barn when the cow goes to the right.
The length of the rope is $16\pi$ units.



The area of the half circle should be no problem. The shape that is made when the cow wraps around is called the involute of a circle. Using some trig/geometry you can parametrize the curve as $x=r\cos t+tr\sin t$ and $y=r\sin t-tr\cos t$. Then use Green's theorem for area $A=\frac{1}{2}\int_C-ydx+xdx.$ This is actually a great problem and can be solved a couple of other ways as well.
If you want to use Riemann sums you will be summing up the area of circular sectors as the angle gets smaller and smaller. Here is a crude picture when $n=4$ and $r$ is the radius of the barn. Taking the limit of the sum as $n\rightarrow\infty$ will get you the answer.
