---
title: "The Trapezium Rule"
description: "Estimate a definite integral by slicing the region into trapezia of equal width, and use the curvature to say whether the estimate is above or below the true value."
canonical: https://lightmysky.com/learn/mathematics/the-trapezium-rule-mt_rAlJ1WIRmZ
source: https://lightmysky.com/learn/mathematics/the-trapezium-rule-mt_rAlJ1WIRmZ.md
retrieved: 2026-09-12
---

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# The Trapezium Rule

Estimate a definite integral by slicing the region into trapezia of equal width, and use the curvature to say whether the estimate is above or below the true value.

Subject: Mathematics · Area: Calculus & Analysis · Ages 17 to 18
Page: https://lightmysky.com/learn/mathematics/the-trapezium-rule-mt_rAlJ1WIRmZ

## Ready when they can

- Estimate an integral with four strips of equal width
- Halve the strip width and describe what happens to the error
- Decide from a sketch whether the estimate is too big or too small

## Lesson: Strips, chords and honest areas

Slice the region under the curve into n strips of equal width h and join the sampled heights with straight tops. Each strip is a trapezium, and adding them gives (h/2) times first plus twice each middle plus last. With n strips you need n + 1 height values, since neighbours share their boundaries.

**Example.** For x squared from 0 to 4 with 4 strips, h = 1 and the heights are 0, 1, 4, 9, 16. The estimate is (1/2) times (0 + 2 times (1 + 4 + 9) + 16) = 22. The exact area is 64/3, about 21.33, so the estimate sits above it.

A sketch predicts the error sign before any arithmetic. Chords of a cup up convex curve sit above the arc, so the trapezia hold too much area and the estimate is too big. For a cap down concave curve the chords sag below and the estimate is too small. Halving the width roughly quarters the error for smooth curves, since the error scales with h squared.

**Tip.** For a straight line or a constant, every chord matches the graph, so the rule is exact. Root x bends cap down, so with 4 strips from 1 to 5 the estimate about 6.76 sits just under the exact 6.79. Only refine while the answer keeps moving.

**Recap.** Join the strip heights with chords, read the bend of the curve for the error sign, and halve the width to shrink the error.

## Practice

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

## Needs first

- [The Definite Integral and the Area Under a Curve](https://lightmysky.com/learn/mathematics/the-definite-integral-and-the-area-under-a-curve-mt_S2IO2S7kgd)

## Opens up

- [Numerical Quadrature: Newton-Cotes and Adaptive Rules](https://lightmysky.com/learn/mathematics/numerical-quadrature-newton-cotes-and-adaptive-rules-mt_fT5kRCUiD5)
- [Riemann Sums and the Definite Integral as a Limit](https://lightmysky.com/learn/mathematics/riemann-sums-and-the-definite-integral-as-a-limit-mt_pwa2SRU6P9)
- [Proof by Deduction and Exhaustion](https://lightmysky.com/learn/mathematics/proof-by-deduction-and-exhaustion-mt_QNWvbkg04f)