Definition: A Riemann sum approximates the area under a curve by adding the areas of thin rectangles whose heights come from the function's values.
What is a Riemann sum?
Split an interval into n pieces of width Δx. On each piece, build a rectangle whose height is f at the left end, the right end, or the midpoint. Add the areas: sum of f(xi) Δx. More rectangles give a better estimate.
How do you compute one?
Worked example: estimate the area under f(x) = x2 on [0, 4] with 4 rectangles, so Δx = 1.
- Left sum: (0 + 1 + 4 + 9)(1) = 14
- Right sum: (1 + 4 + 9 + 16)(1) = 30
The exact area is 64/3, about 21.33, between the two estimates. Because x2 is increasing, the left sum underestimates and the right sum overestimates.
What do students slip on?
Using the wrong endpoints or wrong width. Mathbot computes Δx = (b – a)/n first, lists the x-values, and only then evaluates f. As n grows without bound, the sum becomes the definite integral.