Parametric Calculus: Arc Length: Example 3: Cycloid

In this video I go over another example on determining the arc length of a parametric curve and this time determine the length of one arch of a the famous cycloid shape. In the derivation I use the arc length formula for parametric curves as well as the Pythagorean and Half-Angle Trigonometric Identities. The resulting value is that the arch length of one arch is 8 times the radius of the circle that generates the cycloid! This is quite a fascinating result, and so I have also included a brief history lesson on this finding as well. The length of a cycloid was first found by the English Architect Sir Christopher Wren in 1658. Wren’s method of determining the cycloid length was much more tedious than our more advanced calculus version, in which his method required the dissection of the cycloid into segments of a circle. Along with Wren, I also go over a brief history lesson on the famous St. Paul’s Cathedral, in which Wren was the designer and architect, and bring up the very interesting point that it looks oddly like the United States White House. This similarity is something that I may look further into in the near future… This is a great video in deriving the length of a cycloid, as well as some of the history behind it, so make sure to watch this video!

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Related Videos:

Parametric Calculus: Arc Length: Example 2: Unit Circle: https://youtu.be/X-iElfvx34c Parametric Calculus: Arc Length: Example 1: Unit Circle: https://youtu.be/leyln2bRCpk Parametric Calculus: Arc Length Part 1: https://youtu.be/AWvJDK-m6wQ Parametric Calculus: Areas: https://youtu.be/XdplYV61xlM Parametric Calculus: Tangents: https://youtu.be/deQwD2o0Sas Parametric Equations and Curves: https://youtu.be/Kd3XF4LZoFE Applications of Integrals: Arc Length Proof: https://youtu.be/2rb4H_rmgxg Trigonometry Identities: Proof that sin^2(x) + cos^2(x) = 1: http://youtu.be/o-fAx_96lgw Half Angle Trigonometry Identities: http://youtu.be/0bY6tHZhBSI .

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