Riemann Sums

Calculus 1 · Axiom Academy

Approximate the area under a curve with rectangles, then watch those approximations sharpen into the exact definite integral. Take a continuous function f on an interval [a, b] that stays above the x-axis. We want the area trapped between the curve and the axis. For a line or a circle we have formulas; for a general curve we have nothing — so we need a way to build the area out of pieces we already know how to measure. First we cut the base. Split [a, b] into n equal subintervals; each has the same width . The cut points form a partition , and every strip will sit on one of these subintervals. 3. Choosing the Height: Three Riemann Sums Each rectangle's width is ; its height is the curve's value at one sample point inside the subinterval. Three standard choices give three approximations: Height f(x_ i-1 ) — the left endpoint of each subinterval. Height f(x_i) — the right endpoint of each subinterval. Height — the midpoint of each subinterval. Every version follows the same pattern — add the areas of all n rectangles. Only the sample point changes: where is the midpoint of the i -th subinterval. Each term is literally one rectangle's area. For f(x) = x^2 with , the rectangle sampled at has height f(1.25) = 1.5625 and area . Sum four such midpoint rectangles on [0, 2] and you get M_4 = 2.625 . 5. More Rectangles, Less Error

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