The trapezoid rule approximates area by dividing the interval into strips and using trapezoids. More strips = better approximation. Try n = 1, 2, 4 strips.
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The trapezoid rule: Used to approximate a definite integral when exact integration is difficult. ∫ f(x) dx from a to b ≈ (h/2)(y₀ + 2y₁ + 2y₂ + … + 2yₙ₋₁ + yₙ)
Reading the formula
- h = (b - a)/(n) is the strip width.
- yk = f(a + kh) are the y-values at equally spaced x-values.
- 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 ∫₀² x² dx using the trapezoid rule with n = 4 strips.
- h = (2 − 0)/4 = 0.5. x-values: 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: ∫₀² x² dx = [x³/3] from 0 to 2 = 8/3 ≈ 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.
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Building the table of values: For the trapezoid rule, always start by building a table of values.
| x | y = f(x) |
|---|---|
| x₀ = a | y₀ |
| x₁ = a + h | y₁ |
| ... | ... |
| xₙ = b | yₙ |
Then apply the rule: Then apply: T ≈ (h/2)(y₀ + 2y₁ + 2y₂ +... + 2yₙ₋₁ + yₙ)
Example: n = 5 strips
Step by step
- Approximate ∫₁³ (1/x) dx using n = 5 trapezoid strips.
- h = (3 − 1)/5 = 0.4. x-values: 1, 1.4, 1.8, 2.2, 2.6, 3.0.
- y-values: 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 and the formula: Even when using the GDC, show the table of x and y values and the formula. 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.
Read it from the bend
- 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.
No second derivative needed: 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: say why from the shape: 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, you are often asked to compare it with the exact value.
The formula: Percentage error = |approx − exact| / |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 cross-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 cross-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]
- ∫₀⁴⁰ (0.04x² − 0.001x³) dx = [0.04x³/3 − 0.001x⁴/4] from 0 to 40.
- 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 ≈ 6.25%.
Final answer
T ≈ 200, exact = 213.33, percentage error ≈ 6.25%
Percentage error in three steps: The percentage error calculation is straightforward once you have both values: 1.
Read your trapezoid answer from part (a) 2.
Use the exact integral from part (b) 3.
Apply the formula: |approx − exact| / |exact| × 100% Divide by the exact value, never by the approximate value.
Find exact area using the GDC: On Paper 2 the exact integral is often too hard to do by hand (e.g., it involves cube roots or non-standard functions). Use the GDC's integration function. Still write: 'Area = ∫[a→b] f(x) dx = [value]' to show the GDC was used.