Parametric Calculus: Arc Length Part 3 (Debunking My Own Video)

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In this video I revisit my earlier video Part 3 of the proof of the arc length formula for parametric curves and show that I was WRONG! I had recently learned that the steps I used in Part 3 were completely wrong and so I have decided to make that video UNLISTED (https://youtu.be/jRphcD2qAIY). (In this video now I mentioned I made it private, but decided to make it unlisted instead so that you can view it). My calculus book did not have the derivation for Part 3 so I quickly came up with a derivation and noticed that the result was correct so I overlooked that the steps I took were actually false!

That is why in this video, I debunk my old video, as well as illustrate how the actual derivation is much more straight-forward. In my earlier video, the biggest error that I made was that I assumed that squaring the integral or summation is the same as squaring the integrand or function inside the sum. This is in fact WRONG! We can’t simply do that, and I illustrate this by showing several examples where this is clearly false. Thus it is important to deal with integrals and summations as a whole, instead of trying to dissect them as I had done.

As for the actual derivation, the main proof that I was trying to show was that the summation from Part 2 formed from polygonal approximation had two variables that were not the same. Thus the sum was not exactly a Riemann Sum. But in this video I illustrate how as the number of intervals in the polygonal approximation approaches infinity, the parameter interval decreases to infinitely small, i.e. approaching zero. Also, combined with the fact that the two variables are within this interval, as it decreases in size the variables approach a singular value. Thus we end up approaching in fact a Riemann Sum, and thus we can use this fact to write the sum as an integral, which is the same as in Parts 1 and 2. This is a very important video because it shows the importance to always critically apply each step or mathematical property to ensure that not only the final result is correct, but that the steps are as well, so make sure to watch this video!

Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhuUzBJ6WCdlnK2dkHQ

View Video Notes on Steemit: https://steemit.com/mathematics/@mes/parametric-calculus-arc-length-part-3-debunking-my-own-video