Quadrature rule

Summary

A quadrature rule approximates a definite integral $\int_a^b f$ by a weighted sum $\sum_i w_i f(x_i)$ of values of $f$ at fixed nodes. It is the basic object of numerical integration in one dimension. Rules are compared by the polynomials they integrate exactly and by their error for smooth functions. Newton–Cotes rules use equally spaced nodes, and Gaussian rules use optimally placed nodes.