The usual way of numerically integrate a function, is to find a simpler function which approximates the given function and then integrating the interpolation function. That is, if we want to find , we find an approximating function such that and be close (on some concept of distance) and then we say

The simplest approximation functions are polynomials. If we evaluate at some points , we can use Lagrange's interpolating polynomial to find a polynomial with degree such that for .

Newton and Cotes' integration formulas are obtained when the are sampled evenly over the interval, and then Lagrange interpolating polynomials are used to approximate the function.

The Newton and Cotes formulas for small values of are given on the following table.

		Name
		Trapezoidal rule
		Simpson's rule
		Simpson's 3/8 rule
		Milne's rule

Since the Simpson's rule is actually the Newton and Cotes formula for , the proof of Simpson's rule illustrates this method.