Riemann Sums - Approximating Area Under a Curve

Calculus 1 · Axiom Academy

Approximate the area under a curve with rectangles, then sharpen that approximation until it becomes the definite integral. Take a continuous function f(x) on an interval [a, b] and ask for the area trapped between its curve and the x -axis. Sweeping a line from a to b , the shaded region grows — but there is no single formula for it the way there is for a box. We need a way to measure a curved region using shapes we already understand . Cut [a, b] into n equal pieces. Each piece — each subinterval — has the same width, and the cut points march evenly from a to b . On every subinterval we stand up a rectangle, so the curved area becomes a row of rectangles whose total area is easy to compute. 3. Left, Right, and Midpoint Sums To set a rectangle's height we read the function at one sample point inside each subinterval. Three natural choices give three sums. The animation runs all three on the same curve over [0, 4] with n = 4 , and tallies each total against the true area . 4. Let : the Definite Integral Now turn the only free knob — the number of rectangles. As n climbs from 4 to 8 to 16 to 40 , each strip narrows and the staircase hugs the curve ever tighter. The running total stops drifting and homes in on one value. The animation shows the midpoint sum closing on the exact area . You turned a curved area into a row of rectangles, chose how tall to make them, and watched the sum converge to the definite integral.

This is the written version of the interactive lesson above. See the full Calculus 1 course.