Rectangles

Riemann sums

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• The area under a curve can be approximated by thin rectangles: a Riemann sum.
• As the rectangles get thinner, the sum approaches the exact area, the definite integral.
• $\int_a^b f(x)\,dx$ is the limit of $\sum f(x_i)\,\Delta x$.

Steps

1. Distance from a speed graph
2. Curved tops
3. More rectangles
4. The limit is the exact area
5. Estimate by hand
6. Over or under?