Newton and Cotes formulas

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 ${\int }_a^{b}f(x)𝑑x$, we find an approximating function $p(x)$ such that $f(x)$ and $p(x)$ be close (on some concept of distance) and then we say

$${\int }_a^{b}f(x)𝑑x\approx {\int }_a^{b}p(x)𝑑x$$

The simplest approximation functions are polynomials. If we evaluate $f(x)$ at some points $x_0,x_1,\mathrm{\dots },x_n$, we can use Lagrange’s interpolating polynomial to find a polynomial $p(x)$ with degree $n$ such that $p(x_j)=f(x_j)$ for $j=0,1,\mathrm{\dots },n$.

Newton and Cotes’ integration formulas are obtained when the $x_0,x_1,\mathrm{\dots },x_n$ 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 $n$ are given on the following table.

$n$	$\int p(x)$	Name
$1$	$\frac{h}{2}(f(x_0)+f(x_1))$	Trapezoidal rule
$2$	$\frac{h}{3}(f(x_0)+4f(x_1)+f(x_2))$	Simpson’s rule
$3$	$\frac{3h}{8}(f(x_0)+3f(x_1)+3f(x_3)+f(x_3))$	Simpson’s 3/8 rule
$4$	$\frac{2h}{45}(7f(x_0)+32f(x_1)+12f(x_2)+32f(x_3)+7f(x_4))$	Milne’s rule

recalling that $x_0,x_1,\mathrm{\dots },x_n$ are evenly spaced on $[a,b]$.

Since the Simpson’s rule is actually the Newton and Cotes formula for $n=2$, the proof of Simpson’s rule illustrates this method.