Riemann sums: left, right and midpoint
Problem 4.542 · medium
Compute the midpoint Riemann sum \( \displaystyle M_{4} \) for \( \displaystyle f(x) = 2 x^{2} + 2 x + 7 \) on \( \displaystyle [-1, 0] \).
- \[ 1 \cdot \frac{1}{4} = \frac{1}{4} \]Δx = (b − a)/n.✓ Proved
- The midpoint points are x = -7/8, -5/8, -3/8, -1/8.
- \[ \frac{\frac{209}{32} + \frac{209}{32} + \frac{217}{32} + \frac{217}{32}}{4} = \frac{213}{32} \]Δx times the sum of the function values there.✓ Proved
- For comparison, the exact integral is 20/3 ≈ 6.6667.
Answer \( M_{4} = \frac{213}{32} \approx 6.6562 \)
Lines: 2 proved, 2 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 | Not checked | — | 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: fail (error) — The function values at the midpoints are miscomputed. For x=−3/8 and x=−1/8 the correct value is 218/32, not 217/32, so the resulting Riemann sum is incorrect.qwen3.6:27b-mlx: pass — The solution correctly identifies the subinterval width, the midpoints, and the final sum. The comparison with the exact integral is accurate and helpful context.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — The function values at the midpoints are miscomputed. For x=−3/8 and x=−1/8 the correct value is 218/32, not 217/32, so the resulting Riemann sum is incorrect.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the subinterval width, the midpoints, and the final sum. The comparison with the exact integral is accurate and helpful context.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the subinterval width, the midpoints, and the resulting Riemann sum. The final comparison with the exact integral is accurate.
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 2026-10-06 with SymPy 1.14.0.