Fundamental Theorem, continued

I spent an hour yesterday morning, preparing for my tutoring session with Artemis. That's the first time I've done that. I figured there were 3 questions that followed from what we'd done last week:

1. Why does the sum of the first n squares turn out to be n(n+1)(2n+1)/6? (I tried to think about it with drawings, but got nowhere. I looked up "sum of first n squares visual" and found this discussion at a blog called Understanding. This pdf was linked to, and it's the best thing I found. But Jason Dyer pointed me to this much better visual. Now I get it.)
2. What if we wanted that same shape, but could afford a little extra weight, and wanted to find the area out to a variable right edge, labeled R? That was a way to rehash what we'd done the week before, with a little bit more generality.
3. A car part that's under some crazy function.

 When I explained my three questions to Artemis, he said he didn't want to do any of them. I said I was so excited by this stuff, I could stand to do the first question some other time, but I really wanted to look at the second question. I think he was concerned it would be hard. But once we started, he was very excited about it. We figured out that it would be R^3/3. As we took the limit as n (number of slices) approached infinity, he got a kick out of pointing out the parts that would go to 0.

We haven't done many derivatives yet, so he didn't notice that R^3/3 was the anti-derivative of x^2 (at x=R). I asked him to find the area under a triangle with angled side y=x and right side at x=r. That was easy. I wondered what the area under y=x^3 would be, and he saw the pattern, and guessed it. We haven't proved it yet.

We tried to find the area under y=2^x, but we'd never done derivatives of exponential functions yet, so we weren't able to finish that one.

We have one more session before I go on vacation. Maybe we'll tie up some loose ends, or maybe we'll play with something less strenuous. I'll let him decide.