Year 11 Maths Advanced Module 3 ⏱ ~35 min Lesson 14 of 15

Trapezoidal Rule

Not every function can be integrated exactly. When we only have data points or when the anti-derivative is unknown, we need a reliable way to estimate the area under a curve. The trapezoidal rule replaces the curve with straight-line segments, turning complex areas into simple trapeziums that anyone can calculate.

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Think First

Suppose you want to estimate the area under a curved hill, but you only know the height at a few evenly spaced points along the base. One approach is to connect the height measurements with straight lines, creating a series of slanted-sided shapes. Do you think this would give an overestimate or an underestimate of the true area? What might make the estimate better?

Type your initial response below — you will revisit this at the end of the lesson.

Write your initial response in your book. You will revisit it at the end of the lesson.

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📐

Formula Reference — This Lesson

Trapezoidal rule
$$\int_a^b f(x) \, dx \approx \frac{h}{2}\bigl(y_0 + 2y_1 + 2y_2 + \cdots + 2y_{n-1} + y_n\bigr)$$ where $h = \frac{b-a}{n}$ and $y_k = f(x_k)$
Special case ($n = 2$ strips, 3 points)
$$\int_a^b f(x) \, dx \approx \frac{h}{2}\bigl(y_0 + 2y_1 + y_2\bigr)$$
Key points
First and last ordinates appear once All middle ordinates appear twice More strips → generally better accuracy
Key insight: The trapezoidal rule is exact for linear functions. For curves, it provides an estimate whose error depends on how curved the function is and how many strips are used.
📖 Know

Key Facts

  • The trapezoidal rule formula
  • How to calculate strip width $h$
  • How to apply the rule to tabulated data
💡 Understand

Concepts

  • Why the trapezoidal rule approximates area with trapeziums
  • How increasing the number of strips improves accuracy
  • When the rule gives overestimates versus underestimates
✅ Can Do

Skills

  • Apply the trapezoidal rule to approximate definite integrals
  • Use the rule with given function values or data tables
  • Compare trapezoidal estimates with exact values
  • Determine whether an estimate is too high or too low

Misconceptions to Fix

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Wrong: (a + b)² = a² + b².

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Right: (a + b)² = a² + 2ab + b²; the middle term 2ab is essential and commonly forgotten.

📚 Core Content

Key Terms
DerivativeThe rate of change of a function at a point; the gradient of the tangent.
DifferentiationThe process of finding the derivative of a function.
Stationary PointA point where the derivative equals zero.
Chain RuleA rule for differentiating composite functions: dy/dx = dy/du × du/dx.
Product RuleA rule for differentiating products: d(uv)/dx = u(dv/dx) + v(du/dx).
OptimisationThe reverse process of differentiation; finding the area under a curve.
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What Is the Trapezoidal Rule?

The trapezoidal rule approximates the area under a curve by dividing it into $n$ vertical strips of equal width, then replacing the curve in each strip with a straight line. Each strip becomes a trapezium.

If the interval $[a, b]$ is divided into $n$ strips, the width of each strip is:

$$h = \frac{b - a}{n}$$

Let $y_0, y_1, y_2, \ldots, y_n$ be the function values at the endpoints of the strips. The total approximate area is:

$$\int_a^b f(x) \, dx \approx \frac{h}{2}\bigl(y_0 + 2y_1 + 2y_2 + \cdots + 2y_{n-1} + y_n\bigr)$$

In words: half the strip width, times the sum of the first ordinate, the last ordinate, and twice all the middle ordinates.

Why this matters for environmental science. Hydrologists use the trapezoidal rule to estimate river cross-sectional areas from depth measurements taken at regular intervals. These area estimates feed directly into flow rate calculations, helping predict floods and manage water resources across Australia.
🎯

Accuracy and Over/Underestimation

The trapezoidal rule is exact for straight lines. For curves, the error depends on the concavity:

In both cases, using more strips (smaller $h$) reduces the error and improves the estimate.

🧮 Worked Examples

Worked Example 1 — Basic Trapezoidal Rule

Stepwise
Use the trapezoidal rule with 2 strips to approximate $\int_0^2 x^2 \, dx$.
  1. 1
    Find $h$
    h = \frac{2 - 0}{2} = 1
  2. 2
    Calculate ordinates
    x_0 = 0 \quad \Rightarrow \quad y_0 = 0
    x_1 = 1 \quad \Rightarrow \quad y_1 = 1
    x_2 = 2 \quad \Rightarrow \quad y_2 = 4
  3. 3
    Apply the formula
    \int_0^2 x^2 \, dx \approx \frac{1}{2}\bigl(0 + 2(1) + 4\bigr) = \frac{1}{2}(6) = 3
  4. 4
    Compare with exact value
    \int_0^2 x^2 \, dx = \Bigl[\frac{x^3}{3}\Bigr]_0^2 = \frac{8}{3} \approx 2.667
    The trapezoidal estimate ($3$) is an overestimate because $y = x^2$ is concave up.
✓ Answer $3$ (overestimate; exact = $\frac{8}{3}$)

Worked Example 2 — Tabulated Data

Stepwise
The depth of a river channel (in metres) at 5-metre intervals is given below. Use the trapezoidal rule to estimate the cross-sectional area.
Distance (m)
0
5
10
15
20
Depth (m)
0
2.4
3.1
2.7
0
  1. 1
    Identify values
    h = 5, \quad y_0 = 0, \quad y_1 = 2.4, \quad y_2 = 3.1, \quad y_3 = 2.7, \quad y_4 = 0
  2. 2
    Apply the trapezoidal rule
    A \approx \frac{5}{2}\bigl(0 + 2(2.4) + 2(3.1) + 2(2.7) + 0\bigr)
    = 2.5 \times (0 + 4.8 + 6.2 + 5.4 + 0) = 2.5 \times 16.4 = 41
✓ Answer $41$ m$^2$

Worked Example 3 — Determining Over/Underestimate

Stepwise
Use the trapezoidal rule with 2 strips to estimate $\int_0^2 \sqrt{x} \, dx$, and determine whether it is an overestimate or underestimate.
  1. 1
    Find $h$ and ordinates
    h = 1
    y_0 = \sqrt{0} = 0, \quad y_1 = \sqrt{1} = 1, \quad y_2 = \sqrt{2} \approx 1.414
  2. 2
    Apply the rule
    \approx \frac{1}{2}\bigl(0 + 2(1) + 1.414\bigr) = \frac{1}{2}(3.414) \approx 1.707
  3. 3
    Check concavity
    f(x) = x^{1/2}, \quad f'(x) = \frac{1}{2}x^{-1/2}, \quad f''(x) = -\frac{1}{4}x^{-3/2} < 0
    Since $f''(x) < 0$, the curve is concave down, so the trapezoidal rule gives an underestimate.
✓ Answer $\approx 1.707$ (underestimate)
⚠️

Common Mistakes — Don't Lose Easy Marks

Using the wrong value for $h$
$h$ is the strip width $\frac{b-a}{n}$, not $\frac{b-a}{n-1}$ or the number of points.
✓ Fix: Count the number of strips (intervals), not the number of ordinates.
Forgetting to double the middle ordinates
Only the first and last $y$-values appear once. All others are multiplied by 2.
✓ Fix: Check your bracket — it should be $y_0 + 2y_1 + 2y_2 + \cdots + 2y_{n-1} + y_n$.
Reversing over/underestimate
Concave up means overestimate; concave down means underestimate. Many students reverse these.
✓ Fix: Visualise the straight line sitting above (concave up) or below (concave down) the curve.
Not showing enough decimal places
When exact values involve square roots, rounding too early can lead to significant errors.
✓ Fix: Keep at least 3–4 decimal places during working, and only round the final answer.

📓 Copy Into Your Books

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📏 Formula

  • $\frac{h}{2}(y_0 + 2y_1 + \cdots + 2y_{n-1} + y_n)$

🔢 Strip width

  • $h = \frac{b - a}{n}$

📈 Accuracy

  • Concave up → overestimate
  • Concave down → underestimate
  • More strips → better estimate

⚠️ Watch out

  • Double all middle $y$-values
  • Keep enough decimal places

📝 How are you completing this lesson?

🧪 Activities

🔍 Activity 1 — Calculate

Use the Trapezoidal Rule

Estimate each integral using the specified number of strips.

  1. 1 Estimate $\int_0^4 x^2 \, dx$ using 4 strips.

    Type your answer:

    Answer in your workbook.

    Answer in your workbook

  2. 2 Estimate $\int_1^3 \frac{1}{x} \, dx$ using 2 strips. Give your answer to 3 decimal places.

    Type your answer:

    Answer in your workbook.

    Answer in your workbook

  3. 3 Use the data below to estimate $\int_0^{12} f(x) \, dx$ with 4 strips.

    $x$
    0
    3
    6
    9
    12
    $f(x)$
    1.0
    2.2
    3.5
    4.1
    5.0

    Type your answer:

    Answer in your workbook.

    Answer in your workbook

  4. 4 Estimate $\int_0^2 (x^3 + 1) \, dx$ using 2 strips, then compare with the exact value.

    Type your answer:

    Answer in your workbook.

    Answer in your workbook
🎨 Activity 2 — Interpret

Analyse the Estimate

Use words to explain your reasoning.

  1. 1 For $y = x^2$, explain why the trapezoidal rule with 2 strips gives an overestimate of the area under the curve.

    Type your answer:

    Answer in your workbook.

    Answer in your workbook

  2. 2 A scientist uses 10 strips to estimate a river cross-section and gets 84 m$^2$. A colleague uses 20 strips and gets 82 m$^2$. Which estimate is likely more accurate, and why?

    Type your answer:

    Answer in your workbook.

    Answer in your workbook
Revisit Your Thinking

Earlier you thought about estimating the area under a curved hill using straight-line segments.

The trapezoidal rule formalises exactly this idea: it replaces the curve with straight lines between data points, creating trapeziums whose areas are easy to calculate. Whether the estimate is too high or too low depends on whether the curve bends upwards or downwards. And the more points we use, the closer the straight-line path hugs the curve, giving us better and better estimates. This is why the trapezoidal rule remains one of the most practical tools in applied mathematics and science.

Now revisit your initial response. What did you get right? What has changed in your thinking?

Look back at your initial response in your book. Annotate it with what you now understand differently.

Annotate your initial response in your book
Saved
Interactive: Trapezoidal Rule Approximator
Revisit Your Initial Thinking

Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?

MC

Multiple Choice

5 random questions from a replayable lesson bank — feedback shown immediately

✍️ Short Answer

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Extended Questions

ApplyBand 4

6. Use the trapezoidal rule with 3 strips to estimate $\int_0^3 (x^2 + 1) \, dx$. Show all working, including a table of ordinates. 3 MARKS

Type your answer below:

Answer in your workbook.

✏️ Answer in your workbook
ApplyBand 4

7. The speed of a car (in m/s) at 2-second intervals is recorded below. 4 MARKS

Time $t$ (s)
0
2
4
6
8
10
Speed $v$ (m/s)
0
4.5
7.2
9.0
10.1
11.0

(a) Use the trapezoidal rule to estimate the distance travelled by the car in the first 10 seconds.

(b) Explain why this estimate is likely an underestimate if the car's acceleration is decreasing.

Type your answer below:

Answer in your workbook.

✏️ Answer in your workbook
AnalyseBand 5

8. Consider $f(x) = e^{-x}$ on $[0, 2]$. 4 MARKS

(a) Use the trapezoidal rule with 2 strips to estimate $\int_0^2 e^{-x} \, dx$. Give your answer to 3 decimal places.

(b) Find the exact value of the integral.

(c) Determine whether the trapezoidal estimate is an overestimate or underestimate, and explain why.

Type your answer below:

Answer in your workbook.

✏️ Answer in your workbook

✅ Comprehensive Answers

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🔍 Activity 1 — Calculate Model Answers

1. $h = 1$. Ordinates: $0, 1, 4, 9, 16$. Estimate = $\frac{1}{2}(0 + 2 + 8 + 18 + 16) = 22$. Exact = $\frac{64}{3} \approx 21.33$.

2. $h = 1$. Ordinates: $1, \frac{1}{2}, \frac{1}{3}$. Estimate = $\frac{1}{2}(1 + 1 + \frac{1}{3}) = \frac{7}{6} \approx 1.167$.

3. $h = 3$. Estimate = $\frac{3}{2}(1.0 + 4.4 + 7.0 + 8.2 + 5.0) = \frac{3}{2}(25.6) = 38.4$.

4. $h = 1$. Ordinates: $1, 2, 9$. Estimate = $\frac{1}{2}(1 + 4 + 9) = 7$. Exact = $\bigl[\frac{x^4}{4} + x\bigr]_0^2 = 4 + 2 = 6$. Overestimate because $y = x^3 + 1$ is concave up on $[0, 2]$.

🎨 Activity 2 — Interpret Model Answers

1. $y = x^2$ is concave up ($y'' = 2 > 0$). The straight-line tops of the trapeziums lie above the curve, so they cover more area than the true region underneath.

2. The estimate with 20 strips is likely more accurate because smaller strips mean the straight-line approximation follows the curve more closely, reducing the error.

❓ Multiple Choice

1. B — $h = 1$, ordinates $0, 1, 4$. Estimate = $\frac{1}{2}(0 + 2 + 4) = 3$.

2. C — The formula is $\frac{h}{2}(y_0 + 2y_1 + 2y_2 + \\cdots + 2y_{n-1} + y_n)$, so all middle ordinates are doubled.

3. B — For concave up curves, the straight-line tops of the trapeziums sit above the curve, causing an overestimate.

4. C — $h = 1$, ordinates $0, 1, 4, 9, 16$. Estimate = $\frac{1}{2}(0 + 2 + 8 + 18 + 16) = 22$.

5. B — $h = \frac{b-a}{n}$ where $n$ is the number of strips.

6. A — $y = \\sqrt{x}$ is concave down ($y'' < 0$), so the trapezoidal rule underestimates the area.

7. B — The trapezoidal rule is exact for linear functions because straight-line segments perfectly match a straight line.

📝 Short Answer Model Answers

Q6 (3 marks): $h = 1$. Ordinates: $x = 0 \rightarrow 1$, $x = 1 \rightarrow 2$, $x = 2 \rightarrow 5$. Estimate $= \frac{1}{2}(1 + 4 + 5) = 5$ [2]. The exact value is $\frac{8}{3}$, so the estimate is an overestimate because $y = x^2$ is concave up [1].

Q7 (4 marks): (a) $h = 2$. Distance $\approx \frac{2}{2}(0 + 9.0 + 14.4 + 18.0 + 20.2 + 11.0) = 72.6$ m [2]. (b) The estimate may be an overestimate or underestimate depending on concavity in each interval; overall accuracy improves with more strips [2].

Q8 (4 marks): (a) $h = 1$. Ordinates: $e^0 = 1$, $e^{-1} \approx 0.368$, $e^{-2} \approx 0.135$. Estimate $= \frac{1}{2}(1 + 0.736 + 0.135) \approx 0.936$ [2]. (b) $y = e^{-x}$ is concave up, so the trapezoidal rule overestimates the area [2].

Consolidation Game

Trapezoidal Rule

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