---
title: "Numerical Quadrature: Newton-Cotes and Adaptive Rules"
description: "Derive quadrature rules by integrating an interpolating polynomial, compare their error orders, and refine only where the integrand misbehaves."
canonical: https://lightmysky.com/learn/mathematics/numerical-quadrature-newton-cotes-and-adaptive-rules-mt_fT5kRCUiD5
source: https://lightmysky.com/learn/mathematics/numerical-quadrature-newton-cotes-and-adaptive-rules-mt_fT5kRCUiD5.md
retrieved: 2026-09-12
---

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# Numerical Quadrature: Newton-Cotes and Adaptive Rules

Derive quadrature rules by integrating an interpolating polynomial, compare their error orders, and refine only where the integrand misbehaves.

Subject: Mathematics · Area: Calculus & Analysis · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/numerical-quadrature-newton-cotes-and-adaptive-rules-mt_fT5kRCUiD5

## Ready when they can

- Derive the trapezium and Simpson rules by integrating interpolating polynomials
- Compare error orders and predict how halving the step changes each error
- Explain when an adaptive rule beats a uniform one and how it decides where to refine

## Lesson: Area under the curve, panel by panel

The trapezium rule replaces the function by a straight chord over each panel and integrates the chord. One panel gives the width times the average of the two endpoint heights. Add the panels and the interior heights get counted twice, which is the familiar composite pattern. You derived this rule by integrating the linear interpolant, so it is exact for straight lines.

Simpson upgrades the chord to a parabola through the endpoints and the midpoint. One panel gives the width over 6 times the first height plus 4 times the middle height plus the last height. Because a parabola fit is richer, the rule is exact for cubics too. That is why Simpson often beats the trapezium by a wide margin.

Error orders tell you the price of accuracy. Halving the step divides trapezium error by about 4 and Simpson error by about 16. Those ratios come straight from the powers of h in the error terms. Refine once and compare answers: the improvement ratio shows whether the theory holds. If it does not, the integrand is rougher than you assumed.

**Tip.** Adaptive rules spend evaluations where they matter. They estimate the error on each subinterval and subdivide only the worst ones. A sharp peak gets a fine mesh while calm regions keep a coarse one. This beats uniform refinement when the difficulty sits in one small region. But tiny steps eventually drown in rounding noise, so every strategy needs a stopping rule.

**Recap.** You integrate a chord for the trapezium and a parabola for Simpson, halve the step for errors that shrink by 4 or 16, and refine only the worst panels when trouble is local.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Polynomial Interpolation and Its Error Term](https://lightmysky.com/learn/mathematics/polynomial-interpolation-and-its-error-term-mt_NSnzY9Xodd)
- [The Trapezium Rule](https://lightmysky.com/learn/mathematics/the-trapezium-rule-mt_rAlJ1WIRmZ)