In this video I go over a very interesting example, which is the somewhat famous “cow (or goat or another farmyard animal) grazing area” problem which involves determining the area in which the cow can graze if it is tied to a circular silo. There are several ways of solving this area problem, and in this video I look at Method 1, which involves using the parametric equations of the involute curve, and which were obtained from my last video on Parametric Calculus: Example 1: Involute; so make sure to watch that video! This initial method is actually VERY tedious, so much so that I had to use an online integral calculator in the middle of this video because otherwise this video would be about an hour long. I may do another video going over the integral manually, which involves using integration by parts multiple times, so stay tuned for that video!

The total grazing area is the area between the involute curve and the silo, as well as the area of a large half-circle to the left of the silo. The area between the involute and the silo is obtained by the basic area method of infinite rectangles which sum up as an integral, but an interesting result occurs when calculating the area. Because the involute is not a one-to-one function, we need to break the involute area into two parts, but then these two parts actually sum to exactly the same result if we instead just solved the integral for the whole range of the function, regardless of it being not a one-to-one function! Thus the integral automatically subtracts the excess area as I showed in this video! I may investigate this further to see if I can come up with a more general formula so stay tuned for that too!

While this method of solving the cow grazing area is very tedious, the final result is nonetheless very compact and is 5(π^3)(r2)/6. I will go over a much simpler way of solving this area in a later video, which involves summing up infinite areas of sectors of circle, like in polar coordinates, and thus we don’t need to use parametric equations. Understanding the different ways of solving the same problem is what makes the overall field of mathematics truly remarkable. Thus make sure to watch this video and the ones that follow on how to solve this “grazing area” problem!

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Example

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.

Solution:

Recall from my last video (Parametric Calculus: Example 1: Involute) in which I went over the "involute" of a circle and the resulting parametric equations of the involute curve.

The involute curve only accounts for the area enclosed as the rope is "unwinding" from the circular silo.

Thus we can break down our total grazing area as follows:

This is an amazing result!

This means that simply taking the integral from θ = π to θ = 0, we automatically subtract the excess area due to the function not being a one-to-one function!

I may investigate this further to develop a more general case so # StayTuned!

This integral is very tedious to solve, so I have just calculated it with an online integral calculator.

I may go over this integral manually in a later video so # StayTuned!

http://www.integral-calculator.com/#expr=r%5E2%281%2F2%2Ax%2Asin%282x%29-x%5E2%2Acos%5E2%28x%29&simplify=1

I will go over a much simpler method to solve for the grazing area, and which doesn't involve parametric equations so # StayTuned!!