Area of the field that the cow can graze.

Solution 1:

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.

Solution 2:

Try to do a Riemann sum. For $0<\theta<\pi/2$, consider the strip of land covered as the contact angle moves from $\theta$ to $\theta+d\theta$. Integrate that. Then, for $\pi/2<\theta<\pi$, the end of the rope is coming back west, and the strip of land you measure is the inaccessible part. Integrate that and subtract from the first answer.

When the contact angle is $\theta$, a bound on the region is $$P1(-16\cos\theta+16(\pi-\theta)\sin\theta,16\sin\theta+16(\pi-\theta)\cos\theta)$$ Another bound is on the south side, $$P2(-16\cos\theta+16(\pi-\theta)\sin\theta, -16\sin\theta-16(\pi-\theta)\cos\theta)$$ Then, allowing just a bit more contact on the barn, to $\theta+d\theta$, you get $$P3(-16\cos(\theta+d\theta)+16(\pi-(\theta+d\theta))\sin(\theta+d\theta),16\sin(\theta+d\theta)+16(\pi-(\theta+d\theta))\cos(\theta+d\theta))$$ and similar for P4. There is a region P1P2P4P3 which is a trapezoid, whose height is $P1y$ and whose width is $d\theta$ times $dP1x/d\theta$

Sorry I don't know how to do pictures on this.

Solution 3:

The length of the rope ($16\pi$) is half the circumference of the barn($2\pi r/2=\pi r$).

$\therefore \pi r =16 \pi\Rightarrow r=16$, this is the radius of the barn. Radius of the cow's grazing area is $16\pi$ which is greater than $32$, $\hspace{30 pt}\therefore$ cow's grazing area encompasses the circular barn.

Also, length of the rope=half the circumference of the barn, which means that cow can just reach the rightmost point in your diagram.

The difference of area of circle formed by cow minus the area of barn will give you the answer.