Finding the area of a circle using Riemann Sums

We’ve moved on to chapter 4 of my son’s calculus book -> integrals. The first sections are talking about Riemann sums.

I was a little surprised at the initial difficulty my son had with the examples. I think one of the problems – maybe the main problem – was how many different things you had to keep track of to evaluate these sums. Once we studied the ideas a bit more and saw that the sums could be really be broken down into keeping track of lengths and widths of rectangles, the ideas seemed to make a lot more sense to him.

Last night I decided to show him a Riemann sum that was different than the polynomial examples he’d already worked through. For this one – finding the area of a circle – he knew the area, but the sum was pretty complicated.

Here’s his work finding the sum:

Next we went to the computer

Finally, I thought it would be fun to show him a little surprise -> the Riemann sum that you would use to find the volume of a sphere is actually pretty easy to evaluate by hand.

I think that’s going to be it for specific Riemann sum background work. Looks like the next sections introduce integrals. Excited to dive into this topic!