Trapezium rule

Mathematics· 9709 (2026–2027) Syllabus: 2.5 Integration (Paper 2)· 15 min read

1. Core Trapezium Rule Formula★★☆☆☆⏱ 3 min

📘 Definition

Trapezium Rule for n equal strips

$\int_{a}^{b} f (x) d x \approx T_{n}$

For an integral over interval [a, b] split into n equal width strips, the total approximate area is the sum of the areas of n individual trapezia, each with two parallel vertical sides of length $y_{i}$ and $y_{i + 1}$

T_n = \frac{h}{2} \left( y_0 + 2y_1 + 2y_2 + ... + 2y_{n-1} + y_n \right) \quad \text{where } h = \frac{b-a}{n}

📐 Worked Example

Write out the full trapezium rule expression for $\int_{1}^{5} x^{2} d x$ using 4 equal strips

1
First calculate interval width h:
2
$h = \frac{5-1}{4} = 1$
3
List all y values at x=1, 2, 3, 4, 5:
4
$y_0 = 1^2 = 1, y_1 = 4, y_2 =9, y_3=16, y_4=25$
5
Substitute into the formula, remembering to double all middle terms:
6
$T_4 = \frac{1}{2} \left( 1 + 2(4 +9 +16) +25 \right) = 42$

2. Step-by-Step Application Workflow★★★☆☆⏱ 4 min

Extract the lower bound a, upper bound b, and number of strips n from the question
Calculate interval width h = (b - a)/n and confirm all x points fit evenly between a and b
Evaluate f(x) at every x from a to b in steps of h, rounding to the required number of decimal places
Multiply all intermediate y-values (not first or last) by 2, sum all terms
Multiply the total sum by h/2 to get the final approximate integral

📐 Worked Example

Approximate $\int_{0}^{2} e^{x^{2}} d x$ using 4 strips, giving your answer to 3 significant figures

1
Calculate h = (2-0)/4 = 0.5
2
Evaluate y values: $y_{0} = e^{0} = 1$ , $y_{1} = e^{0.25} \approx 1.2840$ , $y_{2} = e^{1} \approx 2.7183$ , $y_{3} = e^{2.25} \approx 9.4877$ , $y_{4} = e^{4} \approx 54.5982$
3
Sum adjusted terms: $1 + 2 (1.2840 + 2.7183 + 9.4877) + 54.5982 = 1 + 26.98 + 54.5982 = 82.5782$
4
Multiply by h/2 = 0.25: $0.25 \times 82.5782 \approx 20.6$ (3 s.f.)

✓ Quick check

Test your understanding of the workflow:

How many y-values do you need for a trapezium rule calculation with 8 strips?
- 7
- 8
- 9
- 10
Reveal answer
9 —
You always need n+1 points for n strips, so 8+1=9 total y-values.

3. Error Behaviour and Estimate Direction★★★☆☆⏱ 4 min

🔬 Derivation

Goal:

Show which way the trapezium rule errs by comparing each strip's chord with the curve

Starting from:

The top of each strip is a straight chord joining two adjacent points on the curve, so whether that chord sits above or below the curve fixes the error direction

1
Sketch a single strip: its slanted top edge is the chord joining the two adjacent points where the curve is sampled
2
If the curve is concave down ( $f^{''} (x) < 0$ ) it arches above the chord, so the chord lies below the curve; the strip misses the sliver between chord and curve and its area is less than the true area under that segment
3
If the curve is concave up ( $f^{''} (x) > 0$ ) it sags below the chord, so the chord lies above the curve; the strip includes extra area and overshoots the true area under that segment
4
Summing all strips keeps the same direction: chords below the curve give a total below the exact integral, chords above give a total above it

Result:

For $f^{''} (x) > 0$ (concave up) across [a,b] the trapezium rule overestimates; for $f^{''} (x) < 0$ (concave down) it underestimates; for a straight line the chords lie on the curve and the rule is exact

📐 Worked Example

State if the trapezium rule estimate for $\int_{0}^{π} sin x d x$ with 2 strips is an over or under estimate, and verify with exact value

1
Sketch $y = sin x$ on $(0, π)$ : the arch bulges above the straight chords, so every chord lies below the curve and the estimate should be an underestimate. (Supporting check: $f^{'} (x) = cos x$ , $f^{''} (x) = - sin x < 0$ on $0 < x < π$ , i.e. concave down.)
2
Trapezium estimate: $h = π /2$ , $T_{2} = \frac{π}{4} (0 + 2 (1) + 0) = π /2 \approx 1.571$
3
Exact integral: $\int_{0}^{π} sin x d x = 2$ . Since $1.571 < 2$ , the estimate is an underestimate, matching the concave-down sketch.

4. Exam Question Variations★★★★☆⏱ 4 min

Question Type	Mark Allocation	Common Rounding Requirement
Basic 4-strip estimate	4-5 marks	3 significant figures
Percentage error calculation	2-3 marks	1 decimal place
Over/under estimate justification	2 marks	State the sketch or concavity reason

5. Common Pitfalls

Wrong move:

Forgetting to multiply all intermediate y-values by 2 in the formula

Why:

This mistake produces an estimate roughly half the correct size, and is penalised heavily in marking

Correct move:

Write the full formula $\frac{h}{2} (y_{0} + 2 y_{1} + ... + y_{n})$ on your working page before substituting values

Wrong move:

Using n as the number of points instead of number of strips to calculate h

Why:

This gives an incorrect interval width, invalidating the entire estimate

Correct move:

Explicitly write n = number of strips, then confirm h = (b-a)/n before calculating any y values

Wrong move:

Justifying over/under estimate by saying 'the curve is increasing' instead of referencing concavity

Why:

CIE examiners do not award marks for this reasoning, as increasing functions can produce both over and under estimates

Correct move:

Sketch the curve with its chords and state whether the chords lie above it (overestimate) or below it (underestimate); you may support this with the sign of $f^{''} (x)$ across the interval

Wrong move:

Rounding individual y-values too early, leading to large final estimate errors

Why:

Cumulative rounding of intermediate terms can shift your final answer outside the acceptable range for exam marks

Correct move:

Keep all y-values to at least 2 more decimal places than the required final answer before summing

Wrong move:

Calculating percentage error relative to the trapezium estimate instead of the exact value

Why:

This reverses the error ratio and gives a wrong final percentage

Correct move:

Memorise the formula $\frac{E s t ima t e - T r u e}{T r u e} \times 100%$ to avoid swapping terms

6. Quick Reference Cheatsheet

Parameter	Definition	Formula
Number of strips n	Equal width segments between integration bounds	$n = \frac{b - a}{h}$
Trapezium rule estimate	Approximate integral of f(x) from a to b	$\frac{h}{2} (y_{0} + 2 y_{1} + 2 y_{2} + ... + 2 y_{n - 1} + y_{n})$
Percentage error	Relative difference from exact integral value	$\frac{E s t ima t e - T r u e}{T r u e} \times 100%$
Overestimate condition	Trapezium value higher than true integral	$f^{''} (x) > 0$ (concave up): chords lie above the curve
Underestimate condition	Trapezium value lower than true integral	$f^{''} (x) < 0$ (concave down): chords lie below the curve

What's Next

Mastering the trapezium rule is a key stepping stone to the other numerical method examined in Pure 2 (Paper 2): iterative root-finding. The rule is frequently combined with transcendental functions such as exponential and trigonometric curves that have no elementary antiderivatives, so you will see it reappear in later integration questions. You should now practise applying the rule to functions with non-integer interval widths, and cross-check your estimates against exact integral values to build intuition for typical error sizes. Remember that in 9709 the trapezium rule is assessed on Paper 2 only — it does not appear on Paper 1 or Paper 3 — so this foundation makes the rest of the Paper 2 numerical work far easier to master.