Simple derivation of basic quadrature formulas

TL;DR

This paper presents simple, elementary proofs for basic quadrature formulas like midpoint, trapezoidal, and Simpson's rules, including error estimates, suitable for teaching introductory numerical analysis.

Contribution

It offers straightforward proofs for fundamental quadrature formulas and introduces a corrected trapezoidal rule, enhancing understanding and teaching of numerical integration methods.

Findings

Provides simple proofs using integration by parts.

Derives error estimates based on derivatives of the integrand.

Introduces a corrected trapezoidal rule with improved accuracy.

Abstract

Simple proofs of the midpoint, trapezoidal and Simpson's rules are proved for numerical integration on a compact interval. The integrand is assumed to be twice continuously differentiable for the midpoint and trapezoidal rules, and to be four times continuously differentiable for Simpson's rule. Errors are estimated in terms of the uniform norm of second or fourth derivatives of the integrand. The proof uses only integration by parts, applied to the second or fourth derivative of the integrand, multiplied by an appropriate polynomial or piecewise polynomial function. A corrected trapezoidal rule that includes the first derivative of the integrand at the endpoints of the integration interval is also proved in this manner, the coefficient in the error estimate being smaller than for the midpoint and trapezoidal rules. The proofs are suitable for presentation in a calculus or elementary numerical analysis class. Several student projects are suggested.