Riemann sums: left, right and midpoint
Problem 4.698 · medium
Compute the midpoint Riemann sum \( \displaystyle M_{4} \) for \( \displaystyle f(x) = 12 - x^{2} \) on \( \displaystyle [0, 1] \).
- \[ 1 \cdot \frac{1}{4} = \frac{1}{4} \]Δx = (b − a)/n.✓ Proved
- The midpoint points are x = 1/8, 3/8, 5/8, 7/8.Reviewed
- \[ \frac{\frac{719}{64} + \frac{743}{64} + \frac{759}{64} + \frac{767}{64}}{4} = \frac{747}{64} \]Δx times the sum of the function values there.✓ Proved
- For comparison, the exact integral is 35/3 ≈ 11.6667.Reviewed
Answer \( M_{4} = \frac{747}{64} \approx 11.6719 \)
Lines: 2 proved, 2 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the sum recomputed in 30-digit floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the subinterval width, the midpoints, and the resulting Riemann sum. The comparison with the exact integral is accurate and helpful context.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the subinterval width, the midpoints, and the resulting Riemann sum. The comparison with the exact integral is accurate and helpful context.qwen3.6:27b-mlx: pass 2026-10-08gpt-oss:20b: fail (error) 2026-10-08 — The solution incorrectly computes the midpoint Riemann sum: it divides the sum of function values by 4 instead of multiplying by Δx=1/4, leading to an incorrect value of 747/64. The correct sum of f(x_i) is 2948/64, and M_4 = (1/4)*(2948/64) = 2948/256 = 11.515625, not 11.6719.
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/riemann_sums, checked 2026-10-08 with SymPy 1.14.0.