Trapezoidal rule for area

IB Mathematics Applications and Interpretation SL· 12 min read

1. Core Formula and Derivation of the Trapezoidal Rule★★☆☆☆⏱ 10 min

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The trapezoidal rule replaces the curved edge of the region under $f (x)$ with straight line segments between adjacent data points, creating a series of trapezoids whose total area is easy to calculate. Unlike Riemann sums that use rectangles, this method produces a more accurate approximation for most smooth curves.

🔬 Derivation

Goal:

Derive the standard trapezoidal rule formula for n equal sub-intervals over $[a, b]$

Starting from:

The total area under $f (x)$ is split into $n$ adjacent trapezoids, each of width $h$

1
For a single sub-interval between $x_{i}$ and $x_{i + 1}$ , the area of the trapezoid is $\frac{h}{2} \times (f (x_{i}) + f (x_{i + 1}))$
2
Sum the area of all $n$ trapezoids: $T o t a l \approx \sum_{i = 0}^{n - 1} \frac{h}{2} (f (x_{i}) + f (x_{i + 1}))$
3
Expand the sum: the first and last function values appear once, all middle values appear twice

Result:

The final trapezoidal rule formula is $A \approx \frac{h}{2} [f (x_{0}) + 2 f (x_{1}) + 2 f (x_{2}) + ... + 2 f (x_{n - 1}) + f (x_{n})]$

📘 Definition

Trapezoidal Rule Standard Form

$A \approx \frac{h}{2} (f (a) + f (b) + 2 \sum_{i = 1}^{n - 1} f (x_{i}))$

IB MAI SL required form of the formula, where $a$ and $b$ are the lower and upper bounds of integration

📐 Worked Example

Use the trapezoidal rule with $n = 3$ sub-intervals to approximate the area under $f (x) = 2 x + 1$ between $x = 1$ and $x = 4$

1
Calculate sub-interval width: $h = \frac{4 - 1}{3} = 1$
2
List all function values at endpoints: $f (1) = 3$ , $f (2) = 5$ , $f (3) = 7$ , $f (4) = 9$
3
Substitute into the formula: $A \approx \frac{1}{2} [3 + 2 (5) + 2 (7) + 9]$
4
Simplify to get final result: $A \approx 0.5 \times 36 = 18$

2. Accuracy and Effect of Sub-interval Count★★★☆☆⏱ 8 min

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As you increase the number of equal sub-intervals, the width $h$ shrinks, so the straight edges of the trapezoids fit the curved function more closely. This reduces the total approximation error for all non-linear smooth functions.

📐 Worked Example

Compare trapezoidal approximations of $\int_{0}^{2} x^{2} d x$ using $n = 2$ and $n = 4$ sub-intervals, given the exact value is $\frac{8}{3} \approx 2.6667$

1
For $n = 2$ : $h = 1$ , values $f (0) = 0, f (1) = 1, f (2) = 4$ . Approximation = $0.5 * (0 + 2 * 1 + 4) = 3$ , absolute error = 0.3333
2
For $n = 4$ : $h = 0.5$ , values $f (0) = 0, f (0.5) = 0.25, f (1) = 1, f (1.5) = 2.25, f (2) = 4$ . Approximation = 2.75, absolute error = 0.0833
3
Conclusion: Doubling $n$ cuts the absolute error by ~75% for this quadratic curve

✓ Quick check

Test your understanding of trapezoidal accuracy trends:

What happens to trapezoidal approximation error as you increase the number of equal sub-intervals?
- Error always increases
- Error always decreases
- Error stays identical
- Error becomes exactly zero
Reveal answer
Error always decreases —
Thinner trapezoids fit the curve more closely, reducing total approximation error for all smooth functions

3. Contextual Real-World Applications★★★☆☆⏱ 12 min

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IB MAI SL exam questions almost always present the trapezoidal rule in a real-world context, such as estimating the area of a lake, the cross-section of a river, or the total distance travelled from a velocity graph. You will be given pre-measured data points rather than a full function to work with.

📐 Worked Example

A surveyor measures the width of a river at 5m intervals across a 30m stretch, getting values 0m, 7.2m, 9.5m, 10.1m, 8.8m, 5.3m, 0m. Use the trapezoidal rule to estimate the cross-sectional area of the river.

1
Identify $h = 5 m$ , total data points = 7, so number of sub-intervals $n = 6$
2
Substitute into formula: $A r e a \approx \frac{5}{2} \times [0 + 2 * (7.2 + 9.5 + 10.1 + 8.8 + 5.3) + 0]$
3
Sum middle terms: $7.2 + 9.5 + 10.1 + 8.8 + 5.3 = 40.9$
4
Final area: $2.5 \times 81.8 = 204.5 m^{2}$

4. IB Mark Scheme Alignment★★☆☆☆⏱ 6 min

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Write down the value of $h$ explicitly to earn the first method mark
List all function values before substitution to avoid missing the factor of 2 on middle terms
Keep all intermediate values stored on your GDC to avoid rounding errors
Round your final answer to 3 significant figures as required by IB conventions

5. Common Pitfalls

Wrong move:

Forgetting to multiply all middle function values by 2 in the trapezoidal formula

Why:

Students incorrectly use equal weight for all points, leading to a severely overestimated area

Correct move:

Label points $f (x_{0}), f (x_{1}) ... f (x_{n})$ before substitution to clearly separate first/last terms from middle terms

Wrong move:

Calculating $h$ as $(b - a) / n$ but using number of data points instead of number of sub-intervals

Why:

Confusing total data points with sub-interval count leads to an incorrect $h$ value

Correct move:

If you have $k$ total data points, number of sub-intervals $n = k - 1$ , so $h = (b - a) / (k - 1)$

Wrong move:

Using the trapezoidal rule for a curve below the x-axis without adjusting signs

Why:

The formula returns a negative value, but the question asks for total positive geometric area

Correct move:

Take the absolute value of each function value before applying the trapezoidal rule to get total area

Wrong move:

Rounding intermediate values too early in multi-step calculations

Why:

IB mark schemes penalize final answers off by more than 1% due to premature rounding

Correct move:

Store all intermediate values on your GDC, only round your final answer to 3 significant figures

6. Quick Reference Cheatsheet

Parameter	Formula	IB Required Notation
Sub-interval width $h$	$h = \frac{b - a}{n}$	$a$ = lower bound, $b$ = upper bound, $n$ = number of sub-intervals
Trapezoidal rule total area	$A \approx \frac{h}{2} (f (x_{0}) + 2 \sum_{i = 1}^{n - 1} f (x_{i}) + f (x_{n}))$	All middle terms multiplied by 2
Absolute error	$∣ A_{e x a c t} - A_{a pp r o x} ∣$	Reported to 3 significant figures

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.

2024 · 2
Approximate area under quadratic curve
2023 · 1
Find sub-interval count for target error
2022 · 2
Estimate tunnel cross-sectional area

What's Next

Mastering the trapezoidal rule gives you a reliable, high-scoring method for numerical integration questions that appear in almost every IB MAI SL Paper 2. This skill directly supports your work on other numerical calculus topics, including Riemann sums for area approximation and kinematics problems that estimate total displacement from velocity-time graphs. You will also build on this foundation to solve more complex contextual problems involving irregular shape area estimation for surveying, engineering, and environmental science scenarios that are frequently tested in the IB MAI SL assessment.