In this video I go over the same “cow grazing area” problem in which a cow (or goat or other farmyard animal) is tied to a circular silo and we are asked to determine the total area in which the cow can graze around. But this time I go over another method in solving this problem, Method 2, and show break the area under the involute curve of the circle into infinite sectors of a circle, which we can sum up as an integral. This method is actually much easier than in the parametric equations used in Method 1, and saves plenty of time too. This integration method is based on the area formula that I derived in my earlier video for the area of a polar curve. The only difference is that instead of having a direct formula for the involute curve, we can simply project the angle of a sector of a circle perpendicularly towards the involute because we know the length of the rope. This is actually an ingenious method in solving the grazing area problem and shows that complex problems can be solved easily with a thorough understanding of mathematics. The difference in the amount of work required in both Method 1 and Method 2 also show how certain techniques are better suited than others in certain situations, which illustrates the remarkable flexibility of calculus, so make sure to watch both methods!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhvV-EUB2O3mB0v8-SA
View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/video-notes-polar-coordinates-area-example-3-cow-grazing-area-method-2
Parametric Calculus: Areas: Example 2: Cow Grazing Area: Method 1: https://youtu.be/pYf_c_CK5kE
Polar Coordinates: Area Formula: https://youtu.be/wd-EwiKzqa0
Polar Coordinates: Area: Example 1: r = cos 2ϴ: https://youtu.be/6kavQyGcODo
Polar Coordinates: Area: Example 2: Cardioid and Circle: https://youtu.be/vSzxzNn3EiA
Parametric Equations and Polar Coordinates: https://youtu.be/usSors49Gdw
Area of a Sector of a Circle: Proof: https://youtu.be/lglnySF8DyY
Angles - Degrees vs Radians: What are Radians??: http://youtu.be/bOdrVvQWRdE
Parametric Calculus: Example 1: Involute of a Circle: https://youtu.be/NIGw_dlEzQ4 .
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