Riemann sums: left, right and midpoint
Problem 4.696 · medium
Compute the midpoint Riemann sum \( \displaystyle M_{6} \) for \( \displaystyle f(x) = - 3 x^{2} + 3 x + 9 \) on \( \displaystyle [-1, 2] \).
- \[ 1 \cdot \frac{1}{2} = \frac{1}{2} \]Δx = (b − a)/n.✓ Proved
- The midpoint points are x = -3/4, -1/4, 1/4, 3/4, 5/4, 7/4.Reviewed
- \[ \frac{\frac{81}{16} + \frac{81}{16} + \frac{129}{16} + \frac{129}{16} + \frac{153}{16} + \frac{153}{16}}{2} = \frac{363}{16} \]Δx times the sum of the function values there.✓ Proved
- For comparison, the exact integral is 45/2 ≈ 22.5000.Reviewed
Answer \( M_{6} = \frac{363}{16} \approx 22.6875 \)
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.
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.gpt-oss:20b: fail (error) 2026-10-08 — The midpoint values used in step 3 are incorrect; the correct function values are 65/16, 129/16, 153/16, 153/16, 129/16, 81/16, leading to M₆ = 355/16 ≈ 22.1875, not 363/16.qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly calculates the subinterval width, identifies the midpoints, and computes the Riemann sum. The final result matches the stated answer.
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.