In Geometry, we’re beginning a chapter on areas of polygons, and the first lesson is area of a rectangle. Pretty exciting, huh? My students are mostly ninth-graders, with a few tenth-graders, and I thought they might enjoy seeing how the area of a rectangle is used to estimate the area under a curve, i.e. a Riemann Sum.
I used a Geogebra activity created by Alex Kasantsidis to demonstrate a simple Riemann sum. We discussed how the sum of the rectangles can either overestimate or underestimate the area under the parabola, and how we can get a better approximation of the area by increasing the number of rectangles used.
Then, I had my students work through an activity (you can download it here) to estimate the area under the curve y = 12 – x^2 for x = -1 to x = 3 using eight rectangles. After averaging the left-hand and right-hand sums, they came up with 38.5. The actual area is 38.66…, so with only eight rectangles they achieved very good results!
What my students enjoyed even more, though, was the satisfaction of learning calculus-level mathematics. Hopefully, this activity allayed some of the apprehension they might have when they hear the word “calculus”.