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The Trapezium Rule

When an integral cannot be found exactly, the trapezium rule estimates the area by splitting it into strips and approximating each by a trapezium. More strips give a better estimate.

24 min Video by Zeeshan Zamurred Integration

Edexcel A level Maths: 11.9 The Trapezium RuleWatch the full walkthrough before the notes below.

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What you'll be able to do

State the trapezium rule
Find the strip width h
Apply the rule to estimate an integral
Judge whether it over- or under-estimates

The rule

With $n$ strips of width $h = \frac{b - a}{n}$ and ordinates $y_{0}, y_{1}, \dots, y_{n}$ : $\int_{a}^{b} y d x \approx \frac{h}{2} [y_{0} + y_{n} + 2 (y_{1} + \dots + y_{n - 1})]$ .

\int_{a}^{b} y d x \approx \frac{h}{2} [y_{0} + y_{n} + 2 (y_{1} + \dots + y_{n - 1})]

Trapezium rule.

1Number of ordinates = strips + 1.

4 + 1 = 5

ordinates.

Answer5 ordinates (y₀ to y₄).

Tip — The first and last ordinates have weight 1; all the middle ones have weight 2.

Over- or under-estimate

For a curve that is $convex$ (bends upward) the trapezia lie above the curve, so the rule $over$ -estimates; for a concave curve it under-estimates.

Formula recap

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

Strip width.

\frac{h}{2} [y_{0} + y_{n} + 2 (\dots)]

Trapezium rule.

Common mistakes to avoid

Using n ordinates for n strips.

There are n + 1 ordinates for n strips.

Doubling the first and last ordinates.

Only the interior ordinates are doubled.

Key takeaways

h = (b − a)/n; n strips need n+1 ordinates.
Area ≈ (h/2)[y₀ + yₙ + 2(interior)].
Convex curve → over-estimate; concave → under-estimate.

Test yourself

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