Newton–Cotes formulas

Summary

The Newton–Cotes formulas are the quadrature rules obtained by integrating the interpolating polynomial of the integrand at equally spaced nodes. The closed formulas include the endpoints; the trapezoidal rule and Simpson's rule are the first two. They are simple and exact for polynomials of the interpolation degree. High order formulas have negative weights and are unstable, so in practice low order formulas are applied on many small subintervals.