# Trapezoidal rule for area

> IB Mathematics Applications and Interpretation SL · IB MAI SL
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u5-trapezoidal-rule-for-area/

This module covers derivation, formula application, accuracy analysis, and real-world use of the trapezoidal rule for approximating areas under curves, fully aligned to IB MAI SL assessment criteria.

**Prerequisites:** [Definite integrals and geometric interpretation of area under a curve](https://www.owlsprep.com/study/ib-math-ai-sl-u5-definite-integrals-intro/); [2D shape area calculation for trapezoids](https://www.owlsprep.com/study/ib-math-ai-sl-u2-geometry-areas-of-2d-shapes/)

## Learning objectives

- Derive the trapezoidal rule formula for approximating the area under a curve
- Apply the trapezoidal rule with equal-width intervals to calculate approximate definite integrals
- Analyse the effect of increasing sub-interval count on approximation accuracy
- Solve real-world contextual area estimation problems aligned to IB MAI SL criteria

## Core Formula and Derivation of the Trapezoidal Rule

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:** 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: $Total \approx \sum_{i=0}^{n-1} \frac{h}{2}(f(x_i) + f(x_{i+1}))$
3. Expand the sum: the first and last function values appear once, all middle values appear twice

*Conclusion:* The final trapezoidal rule formula is $A \approx \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n) \right]$

**Trapezoidal Rule Standard Form** — IB MAI SL required form of the formula, where $a$ and $b$ are the lower and upper bounds of integration

*Notation:* $A \approx \frac{h}{2} \left( f(a) + f(b) + 2\sum_{i=1}^{n-1} f(x_i) \right)$

**Worked example:** Use the trapezoidal rule with $n=3$ sub-intervals to approximate the area under $f(x) = 2x + 1$ 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: $A \approx 0.5 \times 36 = 18$

*Calculator:* allowed

## Accuracy and Effect of Sub-interval Count

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.

> **Accuracy Trend**
>
> For a quadratic function, doubling the number of sub-intervals reduces the trapezoidal approximation error by approximately 75%

**Worked example:** Compare trapezoidal approximations of $\int_0^2 x^2 dx$ 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

**Check your understanding**

Test your understanding of trapezoidal accuracy trends:

1. 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

   *Why:* Thinner trapezoids fit the curve more closely, reducing total approximation error for all smooth functions

*Calculator:* allowed

## Contextual Real-World Applications

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.

**Exam command terms**

IB exam questions use these standard command terms for this topic, with specific mark requirements:

- **Estimate** — No exact integral calculation is required, only apply the trapezoidal rule to the given data points

- **Show that** — You must write all substituted values in the trapezoidal formula to earn full method marks

- **Calculate the error** — Compare your trapezoidal result to the exact integral value provided in the question

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

*Calculator:* allowed

## IB Mark Scheme Alignment

> **Mark Allocation**
>
> IB awards 1 method mark for correct $h$ calculation, 2 method marks for correct substitution into the trapezoidal formula, and 1 accuracy mark for the final 3-significant-figure answer

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

**Summary**

- Trapezoidal rule approximates area under a curve using adjacent trapezoids
- All middle function values are multiplied by 2 in the standard formula
- Increasing the number of sub-intervals reduces approximation error
- Contextual questions use pre-measured data points rather than full functions

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Forgetting to multiply all middle function values by 2 in the trapezoidal formula
  - Why it fails: Students incorrectly use equal weight for all points, leading to a severely overestimated area
  - Correct: Label points $f(x_0), f(x_1)...f(x_n)$ before substitution to clearly separate first/last terms from middle terms
- **Wrong:** Calculating $h$ as $(b-a)/n$ but using number of data points instead of number of sub-intervals
  - Why it fails: Confusing total data points with sub-interval count leads to an incorrect $h$ value
  - Correct: If you have $k$ total data points, number of sub-intervals $n = k-1$, so $h = (b-a)/(k-1)$
- **Wrong:** Using the trapezoidal rule for a curve below the x-axis without adjusting signs
  - Why it fails: The formula returns a negative value, but the question asks for total positive geometric area
  - Correct: Take the absolute value of each function value before applying the trapezoidal rule to get total area
- **Wrong:** Rounding intermediate values too early in multi-step calculations
  - Why it fails: IB mark schemes penalize final answers off by more than 1% due to premature rounding
  - Correct: Store all intermediate values on your GDC, only round your final answer to 3 significant figures

## 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 | $A \approx \frac{h}{2} \left( f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right)$ | All middle terms multiplied by 2 |
| Absolute error | $\|A_{exact} - A_{approx}\|$ | Reported to 3 significant figures |

## 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.

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