The trapezium rule estimates a definite integral by splitting the area into n strips of equal width h and treating each as a trapezium. With ordinates y₀, y₁, …, yₙ, the estimate is (h/2)[(y₀ + yₙ) + 2(y₁ + y₂ + … + yₙ₋₁)]. It is most useful for functions you cannot integrate exactly, and whether it over- or under-estimates depends on which way the curve bends.
The trapezium rule
- Strip width: h = (b − a)/n
- ∫ y dx from a to b ≈ (h/2)[(y₀ + yₙ) + 2(y₁ + … + yₙ₋₁)]
- n strips need n + 1 ordinates
- Curve bending upwards (convex): over-estimate. Curve bending downwards (concave): under-estimate
How does the trapezium rule work?
Split the area under the curve into n vertical strips of equal width h. Join the tops of each strip with a straight line, so each strip becomes a trapezium, and add their areas. The end ordinates are counted once and every inside ordinate is doubled, because each inside height is shared by two trapezia.
Build a table of x- and y-values before applying the formula. Most lost marks come from misreading the table or dropping the factor of h/2.
Strips and ordinates
Count carefully: n strips need n + 1 ordinates, so 5 ordinates means 4 strips, not 5. Getting h wrong spoils the whole estimate, so work it out from h = (b − a)/n.
When the trapezium rule is needed
The rule is essential for functions with no simple antiderivative, such as e^(−x²) or √(9 − x²), or for data given in a table with no formula. Work in radians for trig functions, and use the correct logarithm: log for base 10, ln for base e.
Over-estimate or under-estimate?
The top of each trapezium is a straight chord joining two points on the curve. If the curve bends upwards (it is convex), the chords lie above it and the rule over-estimates. If the curve bends downwards (concave), the chords lie below it and the rule under-estimates. More strips give a better estimate.
Comparing with the exact value
When an exact value is known, you may be asked for the percentage error: |estimate − exact| / exact × 100%. Questions can also give a table of real data, such as speeds at regular times, where the trapezium rule estimates a distance.
Worked examples
Example 1. Use the trapezium rule with 3 strips to estimate ∫ x² dx from 0 to 3.
- h = 1. The ordinates at x = 0, 1, 2, 3 are 0, 1, 4, 9.
- Ends: 0 + 9 = 9. Inside: 2(1 + 4) = 10.
- Estimate: ½ × (9 + 10) = 9.5.
Answer: 9.5, a slight over-estimate of the exact value 9.
Marks: M1 for the table and h = 1, M1 for the formula, A1 for 9.5.
Example 2. Use the trapezium rule with 4 strips to estimate ∫ √x dx from 1 to 3.
- h = 0.5. Ordinates at x = 1, 1.5, 2, 2.5, 3: 1, 1.2247, 1.4142, 1.5811, 1.7321.
- Ends: 2.7321. Inside: 2(1.2247 + 1.4142 + 1.5811) = 8.4400.
- Estimate: 0.25 × (2.7321 + 8.4400) = 2.793.
Answer: 2.79 (3 s.f.).
Marks: M1 for the table with h = 0.5, A1 for 2.79.
Example 3. Use the trapezium rule with 4 strips to estimate ∫ e^(−x²) dx from 0 to 1.
- h = 0.25. Ordinates: 1, 0.9394, 0.7788, 0.5698, 0.3679.
- Ends: 1.3679. Inside: 2(0.9394 + 0.7788 + 0.5698) = 4.5760.
- Estimate: 0.125 × (1.3679 + 4.5760) = 0.743.
Answer: 0.743 (3 s.f.).
Marks: M1 for the table with h = 0.25, A1 for 0.743.
Example 4. Estimate ∫ √x dx from 0 to 1 with 4 strips, and say whether it is an over- or under-estimate of the exact value ⅔.
- h = 0.25. Ordinates: 0, 0.5, 0.7071, 0.8660, 1.
- Estimate: 0.125 × [(0 + 1) + 2(0.5 + 0.7071 + 0.8660)] = 0.643.
- y = √x bends downwards, so the chords lie below the curve.
Answer: 0.643, an under-estimate of ⅔ ≈ 0.667.
Marks: M1 for the estimate, B1 for under-estimate with the reason.
Example 5. Estimate ∫ 1/x dx from 1 to 5 with 4 strips, and find the percentage error, given the exact value ln 5.
- h = 1. Ordinates: 1, 0.5, 0.3333, 0.25, 0.2.
- Estimate: ½ × [(1 + 0.2) + 2(0.5 + 0.3333 + 0.25)] = 1.683.
- ln 5 ≈ 1.609, so the error is |1.683 − 1.609| / 1.609 × 100% ≈ 4.6%.
Answer: Estimate 1.68; percentage error about 4.6%.
Marks: M1 for the estimate, M1 for the percentage error formula, A1 for 4.6%.
Common mistakes
- Confusing strips and ordinates. n strips have n + 1 ordinates; 5 ordinates means 4 strips.
- Doubling the end ordinates. Only the inside ordinates are doubled.
- Forgetting h/2. The whole bracket is multiplied by half the strip width.
- Using degrees for trig ordinates. Calculus work, including the trapezium rule, uses radians.
- Guessing over or under. Decide from which way the curve bends, and give that reason.
How the trapezium rule is examined
In Edexcel International A-Level the trapezium rule is examined in Pure 2 (WMA12); in Edexcel A-Level (9MA0) it is a second-year topic. A typical question:
- gives a partly completed table of values to finish
- asks for an estimate using all the ordinates, to a stated accuracy
- asks whether the estimate is too large or too small, with a reason
- asks how the estimate could be improved, or for a percentage error
Practise complete trapezium rule questions, with mark schemes, in the Pure 2 integration exam questions. The Pure 2 integration worksheet and the integration revision engine add further practice.
The rule estimates the same quantity as a definite integral, so compare it with an exact value when you can.
Frequently asked questions
What is the trapezium rule formula?
(h/2)[(y₀ + yₙ) + 2(y₁ + y₂ + … + yₙ₋₁)], where h is the strip width and y₀ to yₙ are the ordinates.
How do you know if the trapezium rule over- or under-estimates?
If the curve bends upwards (convex), it over-estimates. If it bends downwards (concave), it under-estimates.
How can you make the trapezium rule more accurate?
Use more strips, so each strip is narrower and the chords follow the curve more closely.