The Trapezium Rule

When an integral can't be found exactly, estimate it: replace the curve by straight lines, add up the trapezia, and use the shape of the curve to say whether the answer is too big or too small.

Cambridge 9709 Paper 2Edexcel YMA01 WMA12Edexcel 9MA0 Paper 1Edexcel 9MA0 Paper 2AQA 7357 Paper 1AQA 7357 Paper 2≈ 35 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Apply the trapezium rule with a given number of strips.
Organise ordinates in a table and avoid arithmetic slips.
Decide whether the estimate is an over- or underestimate from the shape of the curve.
Compare an estimate with an exact value and comment on accuracy.

Pause and think

You can't integrate $e^{x^2}$ with any rule you know. How could you still estimate $\displaystyle\int_0^1e^{x^2}\,dx$ ?

Section 2 of 14

Core idea

Start hereWhen there's no formula

$\int_0^2\sqrt{1+x^2}\,dx$ is the length-related integral of a curve; it has no simple antiderivative. Engineers still need the number. Replace the curve by straight segments and add trapezium areas: the trapezium rule.

The big idea

$\displaystyle\int_a^by\,dx\approx\frac h2\left[y_0+2(y_1+\cdots+y_{n-1})+y_n\right]$ , $h=\frac{b-a}n$ . Over-estimates for curves bending up (convex), under-estimates for curves bending down (concave).

1. Deriving the rule

More strips, better estimate

strips (rounded) = 1

The dashed tops of the trapezia hug the curve more closely as the number of strips grows, and the estimate approaches the exact

\tan^{-1}2\approx1.107

Adding trapeziums

$\text{strip }i:\ \tfrac12h(y_{i-1}+y_i)$
Trapezium: average of parallel sides × width.
$\sum=\tfrac h2[(y_0+y_1)+(y_1+y_2)+\cdots+(y_{n-1}+y_n)]$
Add all strips.
$=\tfrac h2[y_0+2(y_1+\cdots+y_{n-1})+y_n]$
Inner ordinates appear twice.

Worked example $\int_0^2\sqrt{1+x^2}\,dx$ , 4 strips

Estimate to 3 d.p.

Goal: Approximation.
Strategy: $h=0.5$ ; five ordinates.

$h=\tfrac{2-0}4=0.5$
Strip width.
$y=1,\ 1.1180,\ 1.4142,\ 1.8028,\ 2.2361$
$x=0,0.5,1,1.5,2$ .
$\tfrac{0.5}2[1+2(4.3350)+2.2361]=0.25\times11.9061$
Formula.

Answer

\approx2.977

✓ CheckTrue value

2.958

; convex curve so an over-estimate ✓.

Common misconceptionn strips, n+1 ordinates 4 strips need 5

y

-values.

Pause and think

Over or under for $\int_1^3\ln x\,dx$ ?

Summary

Ends once, middles twice, times $\frac h2$ .
Convex over; concave under.

Ordinate: A y-value at a strip edge.

Trapezium rule for y = 1/(1 + x²), four strips

With

h=0.5

\tfrac12(0.5)\{1+2(0.8+0.5+0.3077)+0.2\}\approx1.104

. The exact value is

\tan^{-1}2\approx1.107

. Where the curve bends down the straight tops undershoot; where it bends up they overshoot.

HistoryBabylonian astronomers. Clay tablets from Babylon, written between about 350 and 50 BC, show astronomers working out how far Jupiter moved by finding the area of a trapezium under a graph of its speed. Historians announced this in 2016: a version of the trapezium rule over 1400 years before it was thought to have been used in Europe.

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