The Riemann Sum and the Definite Integral

The definite integral ∫ₐᵇ f(x) dx is defined as the limit of Riemann sums: slice the interval into n pieces, approximate the area of each slice with a rectangle (or trapezoid), and let n grow without bound. Every numerical integration technique in science and engineering — from physics simulations to option pricing — is a sophisticated descendant of this idea.

Choosing a Method

Left and right endpoint rules use the function value at one edge of each slice; they converge slowly and systematically bias the result. The midpoint rule samples the center of each slice, and the trapezoidal rule averages the two edges. Both are second-order accurate: doubling n shrinks their error by a factor of four, versus only two for the one-sided rules.

Measuring the Error

This calculator compares each approximation against the exact integral computed from antiderivatives — b³/3 for x², b⁴/4 for x³, −cos(x) for sin(x), and 2/3·x^(3/2) for √x — so you can watch the approximation converge as n increases and confirm which method wins for smooth curves.