A further generalization of this concept for interpolation with arbitrary degree polynomials are the Newton-Cotes formulas.

The Lagrange basis polynomials can be used in numerical integration to derive the Newton-Cotes formulas.

In the simplest case, Newton-Cotes formulas of even degree are used, where the nodes are evenly spaced in the interval:

Popular methods use one of the Newton-Cotes formulas (like the midpoint rule or Simpson's rule) or Gaussian quadrature.

For one-dimensional integration, quadrature methods such as the trapezoidal rule, Simpson's rule, or Newton-Cotes formulas are known to be efficient if the function is smooth.

Interpolation with polynomials evaluated at equally spaced points in [a, b] yields the Newton-Cotes formulas, of which the rectangle rule and the trapezoidal rule are examples.

The trapezoidal rule is one of a family of formulas for numerical integration called Newton-Cotes formulas, of which the midpoint rule is similar to the trapezoid rule.