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Trapezoidal, midpoint and Simpson's rules

Problem 4.980 · medium

Approximate \( \displaystyle \int_{0}^{2} x^{3} + 1\, dx \) using the trapezoidal rule with \( \displaystyle n = 4 \).
  1. \[ 1 \cdot \frac{1}{2} = \frac{1}{2} \]
    Δx = (b − a)/n.✓ Proved
  2. The trapezoidal rule uses the points x = 0, 1/2, 1, 3/2, 2 with weights 1, 2, 2, 2, 1, all times 1/4.
  3. \[ \frac{1 \cdot 1 + 2 \cdot 9 \cdot \frac{1}{8} + 2 \cdot 2 + 2 \cdot 35 \cdot \frac{1}{8} + 1 \cdot 9}{4} = \frac{25}{4} \]
    Weighted sum of the function values.✓ Proved
  4. \[ \int\limits_{0}^{2} \left(x^{3} + 1\right)\, dx = 6 \]
    The exact value, ≈ 6.000000, for comparison: the error is about 2.50e-01.✓ Proved
Answer \( T_{4} = \frac{25}{4} \approx 6.250000 \)

Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

The full receipt
LineStatusChecked byDetail
1✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
2Not checked—a sentence; read, not computed
3✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
4✓ Provedsympy 1.14.0simplify(lhs - rhs) reduced to 0
answer, a second way✓ Checked independentlysympy 1.14.0 + mpmath 1.3.0the rule implemented separately in floating point

Reviewers

  • gpt-oss:20b: pass
  • qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The weights in the trapezoidal rule formula are multiplied by Δx/2, not Δx. The solution incorrectly states the multiplier is 1/4 (which is Δx) inst
Every verdict on record (4)
  • qwen3.6:27b-mlx: inconclusive 2026-10-12 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The weights in the trapezoidal rule formula are multiplied by Δx/2, not Δx. The solution incorrectly states the multiplier is 1/4 (which is Δx) inst
  • gpt-oss:20b: pass 2026-10-12
  • qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly scales the weights by 1/4 instead of the step size Δx = 1/2. The trapezoidal rule formula is (Δx/2) * [f(x_0) + 2f(x_1) + ... + f(x_n)], which simplifies to multiplying the weighted sum by Δx/2 = 1/4 only if the weights are 1, 2, ..., 1. However, the sentence states the weights are 1, 2, 2, 2, 1 'all times 1/4', implying the final multiplier is 1/4. Let's re-read carefully. Standard formula: T_n = (Δx/2) * (f(x_0) + 2f(x_1) + ... + 2f(x_{n-1}) + f(x_n)). Here Δx = 0.5. So factor is 0.5/2 = 0.25 = 1/4. The weights inside the parenthesis are 1, 2, 2, 2, 1. So the calculation is (1/4) * (1*1 + 2*(9/8) + 2*2 + 2*(35/8) + 1*9). This matches the equation in line 3. Wait, let me check the arithmetic in line 3. f(0)=1, f(0.5)=1.125=9/8, f(1)=2, f(1.5)=3.875=31/8? No, 1.5^3 = 3.375 = 27/8. 27/8 + 1 = 35/8. Correct. f(2)=9. Sum = 1 + 2(9/8) + 2(2) + 2(35/8) + 9 = 10 + 18/8 + 70/8 = 10 + 88/8 = 10 + 11 = 21. 21 * 1/4 = 21/4 = 5.25. The equation says the result is 25/4 = 6.25. The arithmetic in line 3 is wrong. 1 + 9/4 + 4 + 35/4 + 9 = 14 + 44/4 = 14 + 11 = 25. Ah, 2*(9/8) = 9/4. 2*(35/8) = 35/4. Sum = 1 + 2.25 + 4 + 8.75 + 9 = 25. So the sum is 25. Multiplied by 1/4 gives 25/4 = 6.25. The calculation is actually correct. Why did I think it was wrong? Let's re-evaluate. T_4 = (0.5/2) * [f(0) + 2f(0.5) + 2f(1) + 2f(1.5) + f(2)] = 0.25 * [1 + 2(1.125) + 2(2) + 2(3.875) + 9] = 0.25 * [1 + 2.25 + 4 + 7.75 + 9] = 0.25 * [24] = 6. Wait. 2*3.875 = 7.75. 1 + 2.25 + 4 + 7.75 + 9 = 24. 24/4 = 6. The exact integral is 6. The trapezoidal rule for a cubic is not exact? Wait, trapezoidal rule is exact for linear functions. For cubics, it has error. Let's re-calculate f(1.5). 1.5^3 = 3.375. f(1.5) = 4.375 = 35/8. 2 * 35/8 = 35/4 = 8.75. My previous manual sum: 1 + 2.25 + 4 + 8.75 + 9 = 25. 25/4 = 6.25. Okay, so the calculation yields 6.25. Is the formula application correct? Yes. Is the arithmetic correct? Yes. So why did I doubt it? Because I miscalculated 2*3.875 in my head as 7.75 instead of 8.75? 2 * 3 = 6, 2 * 0.875 = 1.75. 6+1.75=7.75. Wait. 3.875 * 2. 3*2=6. 0.875*2 = 1.75. Sum = 7.75. But 35/8 * 2 = 35/4 = 8.75. Where is the discrepancy? 3.875 = 31/8? No. 1.5^3 = 27/8 = 3.375. f(x) = x^3+1. f(1.5) = 3.375 + 1 = 4.375. 4.375 = 35/8. 2 * 4.375 = 8.75. Okay, so 8.75 is correct. My mental check 2*3.875 was using the wrong function value (forgot the +1). So the solution is correct. Verdict: pass.
  • gpt-oss:20b: fail (error) 2026-10-12 — The sentence incorrectly states that all function values are multiplied by 1/4. In the trapezoidal rule with Δx=0.5, the endpoints should be weighted by Δx/2=0.25, while interior points should be weighted by Δx=0.5 (i.e., coefficient 0.5). The subsequent weighted sum uses the wrong weights, leading to an incorrect approximation.

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/numerical_integration, checked 2026-10-12 with SymPy 1.14.0.

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