Trapezoidal, midpoint and Simpson's rules
Problem 4.680 · medium
Approximate \( \displaystyle \int_{1}^{2} \frac{1}{x^{2}}\, dx \) using the Simpson's rule with \( \displaystyle n = 4 \).
- \[ 1 \cdot \frac{1}{4} = \frac{1}{4} \]Δx = (b − a)/n.✓ Proved
- The Simpson's rule uses the points x = 1, 5/4, 3/2, 7/4, 2 with weights 1, 4, 2, 4, 1, all times 1/12.
- \[ \frac{1 \cdot 1 \cdot \frac{1}{4} + 2 \cdot 4 \cdot \frac{1}{9} + 1 \cdot 1 + 4 \cdot 16 \cdot \frac{1}{49} + 4 \cdot 16 \cdot \frac{1}{25}}{12} = \frac{264821}{529200} \]Weighted sum of the function values.✓ Proved
- \[ \int\limits_{1}^{2} \frac{1}{x^{2}}\, dx = \frac{1}{2} \]The exact value, ≈ 0.500000, for comparison: the error is about 4.18e-04.✓ Proved
Answer \( S_{4} = \frac{264821}{529200} \approx 0.500418 \)
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
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the rule implemented separately in floating point |
Reviewers
gpt-oss:20b: fail (error) — The weighted sum in step 3 is incorrect: the function values and their Simpson weights are mismatched (e.g., the first term uses f(2)=1/4 instead of f(1)=1). This leads to an incorrect approximation.qwen3.6:27b-mlx: fail (error) — The solution incorrectly applies the factor 1/12 instead of Δx/3 (which is 1/12) to the weighted sum, but the weights listed (1, 4, 2, 4, 1) are correct for Simpson's Rule. However, the calculation in line 3 is algebraically incorrect relative to the stated formula in line 2. Line 2 states the weights are 1, 4, 2, 4, 1 multiplied by 1/12. Line 3 calculates (1*(1/4) + 2*(4/9) + 1*1 + 4*(16/49) + 4*(16/25))/12. The terms inside the parentheses do not match the function values f(x) = 1/x^2 at the specified points. For example, f(1)=1, f(5/4)=16/25, f(3/2)=4/9, f(7/4)=16/49, f(2)=1/4. The weights should be applied as 1*f(1) + 4*f(5/4) + 2*f(3/2) + 4*f(7/4) + 1*f(2). Line 3 has mixed up the weights and function values (e.g., it uses 2*(4/9) instead of 4*(4/9) or 2*(16/25) etc, and the coefficients 1, 2, 1, 4, 4 do not match 1, 4, 2, 4, 1). Specifically, the term 1*(1/4) corresponds to f(2) with weight 1, but it is placed first. The term 2*(4/9) uses weight 2 on f(3/2), which is correct for the middle term, but the other terms are scrambled. The sum inside the parenthesis in line 3 is not the correct Simpson's sum. The final numerical result 264821/529200 is actually the correct Simpson's approximation, so the algebra in line 3 is likely a typo in the representation of the sum, but as written, the equation claims a specific incorrect sum equals the correct result. This is an error in the derivation steps shown.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution incorrectly applies the factor 1/12 instead of Δx/3 (which is 1/12) to the weighted sum, but the weights listed (1, 4, 2, 4, 1) are correct for Simpson's Rule. However, the calculation in line 3 is algebraically incorrect relative to the stated formula in line 2. Line 2 states the weights are 1, 4, 2, 4, 1 multiplied by 1/12. Line 3 calculates (1*(1/4) + 2*(4/9) + 1*1 + 4*(16/49) + 4*(16/25))/12. The terms inside the parentheses do not match the function values f(x) = 1/x^2 at the specified points. For example, f(1)=1, f(5/4)=16/25, f(3/2)=4/9, f(7/4)=16/49, f(2)=1/4. The weights should be applied as 1*f(1) + 4*f(5/4) + 2*f(3/2) + 4*f(7/4) + 1*f(2). Line 3 has mixed up the weights and function values (e.g., it uses 2*(4/9) instead of 4*(4/9) or 2*(16/25) etc, and the coefficients 1, 2, 1, 4, 4 do not match 1, 4, 2, 4, 1). Specifically, the term 1*(1/4) corresponds to f(2) with weight 1, but it is placed first. The term 2*(4/9) uses weight 2 on f(3/2), which is correct for the middle term, but the other terms are scrambled. The sum inside the parenthesis in line 3 is not the correct Simpson's sum. The final numerical result 264821/529200 is actually the correct Simpson's approximation, so the algebra in line 3 is likely a typo in the representation of the sum, but as written, the equation claims a specific incorrect sum equals the correct result. This is an error in the derivation steps shown.gpt-oss:20b: fail (error) 2026-10-08 — The weighted sum in step 3 is incorrect: the function values and their Simpson weights are mismatched (e.g., the first term uses f(2)=1/4 instead of f(1)=1). This leads to an incorrect approximation.qwen3.6:27b-mlx: inconclusive 2026-10-08 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly applies the weights in the Simpson's rule formula. It multiplies the function values by the weights (1, 4, 2, 4, 1) but thegpt-oss:20b: pass 2026-10-08
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 2026-10-08 with SymPy 1.14.0.