[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).
Example: Trapezoid rule with n = 4
Step by step
- Approximate \int02 x2 dx using the trapezoid rule with n = 4 strip.
- h = (2 − 0)/4 = 0.5. x-value: 0, 0.5, 1, 1.5, 2.
- y₀ = 0² = 0; y₁ = 0.25; y₂ = 1; y₃ = 2.25; y₄ = 4.
- Area ≈ (0.5/2)(0 + 2(0.25) + 2(1) + 2(2.25) + 4).
- = 0.25 × (0 + 0.5 + 2 + 4.5 + 4) = 0.25 × 11 = 2.75.
- Exact value: \int02 x2 dx = [x3/3]02 = 8/3 \approx 2.667.
Final answer
Trapezoid approximation: 2.75 (exact: 2.667)
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)
Example: n = 5 strips
Step by step
- Approximate \int13 (1)/(x) dx using n = 5 trapezoid strip.
- h = (3 − 1)/5 = 0.4. x-value: 1, 1.4, 1.8, 2.2, 2.6, 3.0.
- y-value: 1/1 = 1; 1/1.4 ≈ 0.714; 1/1.8 ≈ 0.556; 1/2.2 ≈ 0.455; 1/2.6 ≈ 0.385; 1/3 ≈ 0.333.
- T ≈ (0.4/2)(1 + 2(0.714) + 2(0.556) + 2(0.455) + 2(0.385) + 0.333).
- = 0.2 × (1 + 1.428 + 1.112 + 0.910 + 0.770 + 0.333) = 0.2 × 5.553 ≈ 1.111.
- Exact value (from the GDC) ≈ 1.099.
Final answer
Trapezoid approximation ≈ 1.111 (exact: 1.099)
Show the table — it earns marks: Even when using the GDC, 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.
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Overestimate vs underestimate — from the shape: Whether the trapezoid rule over- or under-estimates depends on how the curve bends between the points: • Curve bends UPWARD (concave up, ∪ shape) → the straight trapezoid tops sit ABOVE the curve → OVERESTIMATE. • Curve bends DOWNWARD (concave down, ∩ shape) → the trapezoid tops sit BELOW the curve → UNDERESTIMATE. Decide by looking at the graph or a quick sketch — you do not need the second derivative.
Example: over- or underestimate?
For y = x² on [0, 2], does the trapezoid rule give an over- or underestimate?
Step by step
- Sketch or picture y = x². It bends upward (concave up) on this interval.
- So the straight trapezoid tops lie above the curve.
- Therefore the estimate is an OVERESTIMATE.
Final answer
Overestimate — the curve bends upward, so the trapezoids lie above it.
Bends upward (concave up, ∪)
- Trapezoid tops lie above the curve
- Overestimate
- Example: y = x²
Bends downward (concave down, ∩)
- Trapezoid tops lie below the curve
- Underestimate
- Example: y = −x²
Over/underestimate is a 1-mark question: Justify by stating the shape: 'The curve bends upward, so the trapezoids lie above it — an overestimate.' No second derivative needed.
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).
Full IB-style question: trapezoid → exact → % error (2023 style)
Step by step
- A hill's cro -section is modelled by y = 0.04x2 - 0.001x3 for 0 ≤ x ≤ 40.
- Heights at x = 0, 10, 20, 30, 40 are: y = 0, 3, 8, 9, 0.
- (a) Use the trapezoidal rule with h = 10 to approximate the cro -sectional area. [2 marks]
- T = (10/2)(0 + 2(3) + 2(8) + 2(9) + 0) = 5 × (6 + 16 + 18) = 5 × 40 = 200.
- (b) Write down the exact integral and evaluate it. [4 marks]
- \int040(0.04x2 - 0.001x3) dx = [(0.04x3)/(3) - (0.001x4)/(4)]040.
- At x = 40: (0.04 × 64000)/(3) - (0.001 × 2560000)/(4) = (2560)/(3) - 640 = 853.33 - 640 = 213.33.
- (c) Find the percentage error. [2 marks]
- (|200 - 213.33|)/(213.33) × 100 = (13.33)/(213.33) × 100 \approx 6.25\%.
Final answer
T ≈ 200, exact = 213.33, percentage error ≈ 6.25%
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.
Find exact area using the GDC: 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.