The trapezium rule

The trapezium rule estimates the area under a curve, and so the value of a definite integral, by splitting the area into strips and treating each strip as a trapezium. The formula is ∫ y dx ≈ ½h[(y₀ + yₙ) + 2(y₁ + y₂ + … + yₙ₋₁)], where h is the width of each strip. Use it when an integral is hard or impossible to do exactly, or when you only have a table of values. In the US it is called the trapezoidal rule.

The trapezium rule formula

Notice that n strips have n + 1 ordinates. The first and last ordinates, y₀ and yₙ, are used once. All the ones in between are used twice, because each is the shared side of two neighbouring trapezia.

A quick way to remember it: half the width, times the ends plus twice the middles.

Where the formula comes from

Each strip is a trapezium with parallel sides y₀ and y₁ and width h, so its area is ½h(y₀ + y₁). Adding up all the strips:

Every middle ordinate appears in two brackets, which is where the 2 comes from.

How to use the trapezium rule

1. Work out h = (b − a) ÷ n. Careful: n is the number of strips, not the number of ordinates.
2. Make a table of x-values from a to b in steps of h, and work out each y. Keep at least 4 or 5 significant figures.
3. Add the ends, y₀ + yₙ.
4. Add the middles and double them.
5. Multiply the total by ½h.

Example 1: checking against an exact answer

Use the trapezium rule with 4 strips to estimate the integral of x² from 0 to 2.

h = (2 − 0) ÷ 4 = 0.5.

The exact value is 8/3 = 2.667, so the estimate is 3% too big. With 8 strips it improves to 2.6875. More strips always get closer for a smooth curve.

Example 2: an integral you cannot do exactly

Estimate the integral of √(1 + x³) from 0 to 2, using 4 strips.

There is no simple antiderivative for √(1 + x³), which is exactly when the trapezium rule earns its place. h = 0.5.

The integral is about 3.28 to 3 significant figures.

Here x is the input and y = √(1 + x³). The trapezium rule joins neighbouring marked points with straight lines and adds up the four trapezia under them. The curve bends upwards, so each straight line sits slightly above it, which tells you the estimate of 3.28 is a little too high. The Integral Calculator gives a very accurate value to compare with.

Example 3: from a table of data

A cyclist's speed is recorded every 10 seconds. Estimate the distance travelled in the 40 seconds.

Distance is the area under a speed-time graph. h = 10, the ends add to 0 + 10 = 10, and the middles add to 6 + 8.5 + 9.5 = 24.

The cyclist travels about 290 m. With real data there is no formula for the curve at all, so a numerical method like this is the only option.

Overestimate or underestimate?

The trapezium rule replaces the curve with straight lines, called chords. Whether that gives too much or too little depends on which way the curve bends between the ordinates.

At A-level you can check with the second derivative: if d²y/dx² > 0 throughout the interval, the curve is convex and the rule overestimates; if d²y/dx² < 0, it is concave and the rule underestimates. In Example 1, y = x² has d²y/dx² = 2 > 0, which is why 2.75 came out above 8/3.

Example 4: an underestimate

Estimate the integral of sin x from 0 to π with 4 strips. h = π/4.

The exact answer is 2. The sine curve is concave between 0 and π, so the chords lie below it and the estimate is too small.

The integration practice from the Calculus course gives you exact answers to compare your trapezium rule estimates with. Evaluate the integral, then press Mark to check it in the course.

Making the estimate more accurate

• Use more strips. Halving h roughly quarters the error for a smooth curve, as Example 1 shows: 4 strips are out by 0.083, 8 strips by 0.021.
• Keep accuracy in the table. Rounding each y-value to 2 decimal places can cost more than the method itself.
• Watch for steep or sharp sections. Where the curve changes quickly, the straight lines fit badly, so more strips are needed there.

Common mistakes

• Confusing strips and ordinates. 4 strips need 5 y-values. h = (b − a) ÷ number of strips.
• Doubling the ends. Only the middle ordinates are doubled.
• Forgetting the ½ at the front of the formula.
• Using degrees. For trig functions in calculus, set your calculator to radians.
• Rounding too early. Keep 4 or 5 significant figures in the table and round only the final answer.
• Saying over or under without a reason. Refer to the shape of the curve, or the sign of the second derivative.

Is the trapezium rule in the formula booklet?

Yes, for A-level Maths. Both AQA's and Edexcel's booklets, checked in November 2026, give the trapezium rule as ∫ y dx ≈ ½h{(y₀ + yₙ) + 2(y₁ + y₂ + … + yₙ₋₁)}, where h = (b − a) ÷ n. They give the Newton-Raphson method next to it under numerical methods. You still need to set up the table, choose h correctly and explain whether the answer is an overestimate or an underestimate. At GCSE Higher, the same idea appears as estimating the area under a graph by splitting it into trapezia, for example to estimate a distance from a velocity-time graph.

Practise integration in a short timed paper

A-level Maths: Calculus gives you a short timed paper every day, with 8 questions worth 25 to 35 marks across differentiation and integration. In the US, the same practice opens as AP Calculus AB. The Calculus course shows rectangles and strips building up the area under a curve before moving to exact integration.

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