[Diagram: math-trapezoid-rule] - Available in full study mode
The trapezoid rule: Used to approximate a definite integral when exact integration is difficult. \intab f(x) dx \approx (h)/(2)(y0 + 2y1 + 2y2 + \ldots + 2yn-1 + yn) where h = (b - a)/(n) is the strip width and yk = f(a + kh) are the y-values at equally spaced x-value. The first and last y-values appear once; all middle y-values appear twice (multiplied by 2).
Why the approximation is not exact: The trapezoid rule uses straight lines to approximate the curve. If the curve is concave up (bending upward), the trapezoids overestimate. If the curve is concave down (bending downward), they underestimate. For x², which is concave up, the trapezoid rule gives an overestimate.
Building the table of values: For the trapezoid rule, always start by building a table: | x | y = f(x) | |---|----------| | x₀ = a | y₀ | | x₁ = a + h | y₁ | |... |... | | xₙ = b | yₙ | Then apply: T ≈ (h/2)(y₀ + 2y₁ + 2y₂ +... + 2yn-1 + yₙ)
Show the table — it earns marks: Even on Paper 2 (GDC allowed), IB expects you to show the table of x and y value. IB awards a mark for showing the correct formula or table setup. Do not just write the final answer — show the table, the formula, and the computation.
Worked example
Apply the key method from The Trapezoid Rule for Estimating Areas in a typical IB-style question.
Step by step
- Write the relevant formula or rule first.
- Substitute values carefully and show each step.
- State the final answer with correct units/context.
Final answer
Clear method and context-based interpretation secure most marks.
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Overestimate vs underestimate: The trapezoid rule approximation accuracy depends on the shape of the curve: • Concave UP (f'' > 0) → trapezoids lie ABOVE the curve → OVERESTIMATE • Concave DOWN (f'' < 0) → trapezoids lie BELOW the curve → UNDERESTIMATE To check: compute f''(x) and determine its sign in the interval.
Concave Up (∪)
- f''(x) > 0
- Trapezoids above curve
- Overestimate
- Example: eˣ, x²
Concave Down (∩)
- f''(x) < 0
- Trapezoids below curve
- Underestimate
- Example: −x², ln(x)
Over/underestimate is a 1-mark question: IB regularly ask: 'State whether your estimate is an overestimate or underestimate. Justify.' Justification = state the concavity and why that leads to your answer. Example: 'Overestimate, because f is concave up so the trapezoids lie above the curve.'
Worked example
Apply the key method from The Trapezoid Rule for Estimating Areas in a typical IB-style question.
Step by step
- Write the relevant formula or rule first.
- Substitute values carefully and show each step.
- State the final answer with correct units/context.
Final answer
Clear method and context-based interpretation secure most marks.
Percentage error in the trapezoid rule: After finding the trapezoid estimate, IB always asks you to compare it to the exact value: \text(Percentage error) = (|\text(approx) - \text(exact)|)(|\text(exact)|) × 100\% The 'exact' value is found by evaluating the definite integral (by hand on Paper 1, or by GDC on Paper 2).
Percentage error is worth 2 marks — easy marks: The percentage error calculation is straightforward once you have both value: 1. Read your trapezoid answer from part (a) 2. Use the exact integral from part (b) 3. Apply the formula: |approx − exact| / |exact| × 100% You will lose marks if you divide by the approximate value instead of the exact value.
Paper 2: find exact area by GDC: On Paper 2, the exact integral is usually too complex to evaluate by hand (e.g., involves cube roots or non-standard functions). You are expected to use the GDC integration function. Still write: 'Area = ∫[a→b] f(x) dx = [value]' to show the GDC was used.