Numerical Integration

Integrals appear regularly in many branches of physics and engineering. The underlying concept in numerical integration is essentially to approximate the integral of a function by means of the integration of its corresponding interpolant. This chapter presents the formulas of particular cases of a general family of quadrature rules, usually known as Newton–Cotes formulas. It examines the accuracy of quadrature formulas arising from equispaced interpolation. The chapter presents two definitions, one of the quadrature error and the other of the degree of exactness of an interpolatory quadrature formula. It reviews an important result from the theory of integration, and then illustrates how to obtain this quantitative measurement of the error in a simple case, namely, the midpoint quadrature rule. In general, the mean value theorem for integrals can be used to measure the quadrature error of higher order quadrature formulas. The chapter also presents optimal strategies to integrate periodic functions.