Riemann Sums Review

Numerical Analysis · Axiom Academy

The foundation of numerical integration Riemann sums approximate integrals by summing rectangles. The height can be evaluated at different points: As n increases, all three methods converge to the true integral. But at what rate? Riemann sums are the foundation of all numerical integration. Better methods like the Trapezoid Rule and Simpson's Rule are simply weighted combinations of these basic ideas: The trapezoid rule averages the left and right Riemann sums—and this simple combination has much better accuracy! Three basic rules: Left, Right, and Midpoint evaluate rectangles at different positions. All converge: As n → ∞, all methods approach the true integral. Midpoint wins: For smooth functions, midpoint is often more accurate than left or right. Foundation: Higher-order methods combine these ideas with weighted averages.

This is the written version of the interactive lesson above. See the full Numerical Analysis course.