Tuesday, August 16, 2022

Prepping for Fall, Calc II: Lovely Arc Length Example

I'll be teaching Calc II for the first time in a few years. This is my first time starting out online with it. So I'm preparing my Canvas shell and thinking about how I want to explain each topic in Canvas. (I know the material well enough that I didn't have to prep this much when we were in person.) The extra prep before we start is so much work, but today it feels totally worthwhile.

For arc length I was excited to use "crinkle crankle walls" as an example. Isn't that a pretty wall? And you can actually use fewer bricks this way than for a straight wall, because one layer of bricks here is stronger than it would be straight (so the straight wall would need extra bricks for support). I'm thinking we'll try to prove that assertion in my Calc II class.

It turns out that arc length uses an integral which often has no "elementary solution", meaning there is no anti-derivative using the functions we are familiar with. 
The arc length for y=sin x is...

And this has no "elementary solution".
I often tell my students that we study infinite series to solve the integrals with no easier solution, but I just realized that that won't work here. (Can't do a square root of an infinite series!) 
Ok, no problem. I'm also teaching numerical integration. So I made a google sheet to do Simpson's method, and it turns out beautifully!! (Beautifully meaning that my answer matched the answers on Math SE that people explained in ways that were above my head. I don't know a thing about "elliptical integrals".)
I still need to remember how to explain Simpson's rule, but I'll get that back easily enough. 
If this wall follows a sine wave, then for 6.28 feet (2π feet) of straight distance covered, it has a length of 7.64 feet. That's just over 20% extra length. (Now to think with my students about whether that's better than the straight wall with supports.)

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