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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Love this post. I first encountered this type of wall when visiting University of Virginia, but did not know about the term “Crinkle Crankle Wall,” or the mathematical reasons for its geometric material resource efficiency relative to a straight wall. It is one thing to prove that a sinusoidal wall is stronger than a straight wall using the same amount of material, but how do we prove that a sinusoidal wall is more efficient than almost any other conceivable smooth shape? (E.g. imagine a wall composed of parabolic or elliptical or any other shape you can think of segments joined together, with the proviso that the functions describing the shapes be reasonably “smooth”.). That would be a problem in the Calculus of Variations, which is beyond the curriculum of second year calculus. (Operationally, theorems in Calculus of Variations generally assume that “smooth” functions are those that are “sufficiently” differentiable, e.g., being twice continuously differentiable, three times differentiable, etc..) https://www.library.virginia.edu/news/2022/uva-walking-tour-enslaved-african-americans-at-uva
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