Trapezium rule

Mathematics· 9709 Unit 2: 2.10· 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) 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 why trapezium rule overestimates for concave down curves

Starting from:

A concave down curve lies above the chord connecting any two points on its surface

1
Each trapezium side is a chord connecting two adjacent points on the curve
2
If the curve is concave down, the chord lies below the curve, so the trapezium area is larger than the true curved area under the segment
3
Summing all segments gives a total estimate larger than the exact integral

Result:

For $f^{''} (x) < 0$ across [a,b], trapezium rule overestimates; for $f^{''} (x) > 0$ , it underestimates

📐 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
Calculate second derivative: $f^{'} (x) = cos x$ , $f^{''} (x) = - sin x$ which is negative for all $0 < x < π$ , so curve is concave down, estimate is overestimate
2
Trapezium estimate: h = $π /2$ , $T_{2} = \frac{π}{4} (0 + 2 (1) + 0) = π /2 \approx 1.571$
3
Exact integral: $\int_{0}^{π} sin x d x = 2$ , so estimate 1.571 < 2, which is an underestimate? Wait no: wait $f^{''} (x) = - sin x < 0$ , chord is below curve, trapezium area is smaller than true area, so yes underestimate, matches calculation.

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	Reference concavity explicitly

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:

Calculate the sign of the second derivative across the full interval to justify your conclusion

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 down) across interval
Underestimate condition	Trapezium value lower than true integral	$f^{''} (x) > 0$ (concave up) across interval

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · 12
4-strip trapezium estimate
2022 · 13
Percentage error calculation
2021 · 11
Over/under estimate justification

What's Next

Mastering the trapezium rule is a critical stepping stone for further numerical methods content in Pure 2 and Pure 3, including more accurate Simpson's rule and iterative root finding techniques. This skill 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 and differential equations questions. You should now practice 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. This foundation will make more advanced numerical approximation topics far easier to master in your exam preparation.