Riemann sums, step by step
A Riemann sum is a pile of rectangles whose total area is trying to become an integral. Walk the five steps, then slide n until the error meter goes dark.
01
Name the integral
Write the definite integral first: the function, the lower bound, the upper bound. The sum is an approximation of that number, not a replacement for it.
02
Pick a method
Left and right use an endpoint of each subinterval. Midpoint uses the center. Trapezoid averages left and right. Midpoint and trapezoid usually close in faster.
03
Choose n, then double it
Start small enough to see the rectangles. Then double n. For left and right, the error roughly halves. For midpoint and trapezoid, it falls faster than that.
04
Read the signed error
The meter is the gap between the sum and the exact integral. If the function is increasing, a left sum undershoots and a right sum overshoots. That sign is the teaching, not a bug.
05
Stop when the readout is the integral
The definition of the definite integral is this limit as n grows. Once the error reads as zero to display precision, you have watched the definition happen.
Full bar is the worst single-rectangle error for this function; empty means the sum is within 0.0001 of the exact integral.
Visualizer page: Riemann sum slider. Night-before list: AB cram sheet.