textbook-theory · English
Difference Equations to Differential Equations: Section 4.2 — Numerical Approximations of Definite Integrals
Section 4.2 of Dan Sloughter's open calculus text presents four numerical methods for approximating definite integrals: the left-hand and right-hand rules (error O(1/n)), the trapezoidal rule and midpoint rule (error O(1/n²)), and Simpson's rule (error O(1/n⁴)). The derivations use the Mean Value Theorem and Taylor-polynomial arguments to bound errors rigorously, and worked examples illustrate convergence rates with tabulated approximations for benchmark integrals.
numerical integrationtrapezoidal ruleSimpson's rulemidpoint rule