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.)

Technology Woes and Cheers: Venn diagram edition

I'm writing questions for my Discrete Math course that will be available to my students (and others) through MyOpenMath, a free online homework system. I'm not very good at programming in their environment, but I'm learning. The cool thing about MyOpenMath is that it uses random numbers in the questions so that each student might get a (slightly) different question.

I wanted a way to ask, for a random Venn diagram: What is the set notation for this?

First, I needed a way to make lots of Venn diagrams, all pretty, and all in the same style. I searched the internet for a free online Venn diagram maker. Nothing right showed up. I looked at over a dozen sites. Many wanted me to sign in. That should not be necessary and I skipped those. None of the others were even close to what I wanted, which is pretty simple. Really?! Isn't this something lots of people would want? 

I asked about it on Math Educators Stack Exchange. Within hours, Cameron Williams posted an answer. He made it on desmos for me. (How sweet is that?! Amazing.) I know desmos well, so I was able to modify his version to be exactly what I wanted, in less time than I had already spent searching. (I suggest you go play with it - it's lovely.) And, if you want orange shading instead of blue, it's very easy to modify this to get exactly what you want.

Then I made 17 screenshots of various combinations of the basic regions, and named them based on the set notation. So "(A un B) int not C.jpg" is the filename for ...

 

Next I went back to MyOpenMath, and wrote most of my multiple choice problem. I'm still stuck on how to get it to display a randomly chosen image file. I think the folks at the help forum there will help me out on that. Once I finish fixing it, I'll edit this post to show the question. MyOpenMath allows attached videos to explain how to answer the questions. I think I might do a video for this one. 

So if you want a free online Venn diagram maker, it's here.  I don't know how to help google move this up in the searches so people can find it. Do you?