Newton-Cotes Framework

The Newton-Cotes formulas provide a systematic way to approximate a definite integral by replacing the integrand with a polynomial that passes through a set of equally spaced nodes. Because polynomials integrate exactly and in closed form, this replacement converts the original integral into a weighted sum of function values that is easy to compute.

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The Core Idea

We want to evaluate:

$I(f) = \int_a^b f(x)\,dx$

Instead of integrating $f(x)$ directly, we approximate it with the degree-$n$ Lagrange interpolating polynomial $P_n(x)$ built from $n+1$ equally spaced nodes $x_0, x_1, \ldots, x_n$ on $[a, b]$, and integrate that polynomial:

$I_n(f) = \int_a^b P_n(x)\,dx$

The shaded region is the integral of $P_n(x)$, and its value equals the weighted sum $\sum_k \sigma_k f(x_k)$ where each weight $\sigma_k$ is the integral of the corresponding Lagrange basis polynomial $l_k(x)$.

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Deriving the Weighted Sum

The Lagrange interpolating polynomial can be written as:

$P_n(x) = \sum_{k=0}^{n} l_k(x)\,f(x_k)$

where each $l_k(x)$ is the $k$-th Lagrange basis polynomial. Substituting into the integral:

$I_n(f) = \int_a^b \sum_{k=0}^{n} l_k(x)\,f(x_k)\,dx$

Since the function values $f(x_k)$ are constants with respect to $x$, they can be pulled outside each integral:

$I_n(f) = \sum_{k=0}^{n} f(x_k) \int_a^b l_k(x)\,dx$

This gives us the fundamental Newton-Cotes formula:

$\boxed{I_n(f) = \sum_{k=0}^{n} \sigma_k\, f(x_k)}$

where the Newton-Cotes weights are defined as:

$\sigma_k = \int_a^b l_k(x)\,dx$

Each weight $\sigma_k$ depends only on the positions of the nodes, not on the function values. This means the weights can be computed once and reused for any function on the same interval.

Note

The weights $\sigma_k$ are the integrals of the Lagrange basis polynomials. Geometrically, each $\sigma_k$ represents the signed area under the $k$-th basis function over $[a, b]$. The sum of all weights always equals $b - a$, which follows from the fact that integrating the constant function $f(x) = 1$ exactly gives $b - a$.

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Closed and Open Forms

Newton-Cotes formulas come in two variants depending on whether the endpoints $a$ and $b$ are used as nodes.

Closed Newton-Cotes

The closed form includes both endpoints as nodes. With $n+1$ nodes (including $a$ and $b$), the step size between consecutive nodes is:

$h = \frac{b - a}{n}$

The nodes are:

$x_k = a + k h, \qquad k = 0, 1, \ldots, n$

so that $x_0 = a$ and $x_n = b$. The two most commonly used closed formulas are:

$n$	Rule	Number of nodes
1	Trapezoidal Rule	2
2	Simpson’s Rule	3

Open Newton-Cotes

The open form excludes both endpoints. The $n+1$ interior nodes are shifted inward by one step from each endpoint, giving a step size of:

$h = \frac{b - a}{n + 2}$

The nodes are:

$x_k = a + (k + 1)h, \qquad k = 0, 1, \ldots, n$

Open formulas are useful when $f(x)$ cannot be evaluated at the endpoints, for example when $f(a)$ or $f(b)$ is undefined.

Note

The most elementary open formula ($n = 0$) uses a single node at the midpoint $x_0 = \frac{a+b}{2}$, giving the Midpoint Rule: $I_0 = (b-a)\,f\!\left(\frac{a+b}{2}\right)$. Despite using only one function evaluation, the Midpoint Rule is $O(h^2)$ accurate, the same order as the Trapezoidal Rule, because the midpoint cancels the leading error term by symmetry.

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Summary

The entire Newton-Cotes framework reduces to three steps:

1. Choose $n+1$ equally spaced nodes on $[a, b]$.
2. Compute the weights $\sigma_k = \int_a^b l_k(x)\,dx$ for each node.
3. Evaluate $I_n(f) = \sum_{k=0}^{n} \sigma_k f(x_k)$.

The next two sections carry out this programme in full for $n = 1$ (Trapezoidal Rule) and $n = 2$ (Simpson’s Rule).