Boole's rule¶

A closed Newton–Cotes quadrature rule using five equally spaced samples to approximate an integral by a weighted quartic interpolant.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3510
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Numerical Analysis → Mathematics

Core Idea¶

Boole’s rule approximates the integral over four equal subintervals by (2h/45)[7f(x0)+32f(x1)+12f(x2)+32f(x3)+7f(x4)], with a composite form applied panelwise. The Lagrange interpolating polynomial is integrated exactly, producing fixed symmetric weights and an error term governed by a higher derivative under smoothness assumptions. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

How would you explain it like I'm…

Five-Spot Hill Measuring

Imagine you want to know how much paint it takes to fill the space under a bumpy hill drawing. Boole's rule says: measure how tall the hill is at five spots spaced evenly apart, then add those heights up using a special recipe where the ends count a little, the in-between spots count a lot, and the very middle counts in between. That recipe gives a very close answer for smooth hills.

The 7-32-12-32-7 Recipe

Boole's rule is a way to estimate the area under a curve without doing the exact math. You split the stretch you care about into four equal pieces, which gives five evenly spaced points. You check the curve's height at each point and combine them with fixed weights: 7, 32, 12, 32, 7, then multiply by a small factor based on the spacing. The rule works really well when the curve is smooth, and for a long stretch you just repeat it on one block of four pieces after another.

Five-Point Weighted Integration

Boole's rule is a numerical integration formula: it approximates the integral of f over an interval cut into four equal subintervals of width h, using the five endpoints x0 through x4. The estimate is (2h/45)[7f(x0) + 32f(x1) + 12f(x2) + 32f(x3) + 7f(x4)]. The weights come from fitting the unique polynomial through those five points and integrating that polynomial exactly. Compared with simpler rules like the trapezoid rule, it uses more points per panel to get a much smaller error, but only if the function is smooth enough that its higher derivatives are well behaved. If the points are not equally spaced, or you change the step convention, those weights no longer apply.

Boole's rule is a closed Newton–Cotes quadrature formula on five equally spaced nodes x0,…,x4 spanning four subintervals of step h. It approximates the integral from x0 to x4 by (2h/45)[7f(x0) + 32f(x1) + 12f(x2) + 32f(x3) + 7f(x4)]. The weights are obtained by integrating the Lagrange interpolating polynomial through the five nodes exactly, which is why they are fixed and symmetric. The error term is governed by a higher derivative of f, so the claimed order of accuracy depends on smoothness assumptions holding on the interval. The composite version partitions a longer interval into panels of four subintervals each and applies the formula panelwise. Correct use requires checking that the node count and spacing, the step convention h, and the weights all match the formula, and that the remainder assumptions are met. Swapping in unequal nodes or a different step convention yields a different rule, not Boole's rule.

Scope of Application¶

Boole's rule belongs to numerical analysis and is useful where the analyst can specify the typed numerical analysis carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate nodes are five equally spaced points spanning four subintervals, weights and step convention match the formula, and smoothness and remainder assumptions support the claimed order. The scope is broad within that domain but bounded by the need for nodes are five equally spaced points spanning four subintervals, weights and step convention match the formula, and smoothness and remainder assumptions support the claimed order. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making nodes are five equally spaced points spanning four subintervals, weights and step convention match the formula, and smoothness and remainder assumptions support the claimed order the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Boole's rule can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Boole's rule. Boole's rule compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed numerical analysis carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express nodes are five equally spaced points spanning four subintervals, weights and step convention match the formula, and smoothness and remainder assumptions support the claimed order independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of numerical analysis because they reuse the typed numerical analysis carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, The Lagrange interpolating polynomial is integrated exactly, producing fixed symmetric weights and an error term governed by a higher derivative under smoothness assumptions., and type the carrier, state every parameter and convention in the definition, test that nodes are five equally spaced points spanning four subintervals, weights and step convention match the formula, and smoothness and remainder assumptions support the claimed order, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Boole's rule Domain-specific

Parents (1) — outgoing typed relations

Boole's rule is a kind of Approximation Prime

What makes it its own entry: the domain-specific identity determined by nodes are five equally spaced points spanning four subintervals, weights and step convention match the formula, and smoothness and remainder assumptions support the claimed order.

Hierarchy path (1) — routes to 1 parentless root

Boole's rule → is a kind of Approximation → is a decomposition of Representation → presupposes Abstraction

Neighborhood in Abstraction Space¶

Boole's rule sits in a crowded region of the domain-specific corpus (15^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Function Analysis & Numerical Methods (51 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25