Riemann sums: left, right and midpoint
Problem 4.701 · medium
Compute the midpoint Riemann sum \( \displaystyle M_{4} \) for \( \displaystyle f(x) = 12 - x^{2} \) on \( \displaystyle [1, 2] \).
- \[ 1 \cdot \frac{1}{4} = \frac{1}{4} \]Δx = (b − a)/n.✓ Proved
- The midpoint points are x = 9/8, 11/8, 13/8, 15/8.
- \[ \frac{\frac{543}{64} + \frac{599}{64} + \frac{647}{64} + \frac{687}{64}}{4} = \frac{619}{64} \]Δx times the sum of the function values there.✓ Proved
- For comparison, the exact integral is 29/3 ≈ 9.6667.
Answer \( M_{4} = \frac{619}{64} \approx 9.6719 \)
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 solution miscomputes the function values at the midpoints and consequently the Riemann sum. The correct values are f(9/8)=687/64, f(11/8)=647/64, f(13/8)=599/64, f(15/8)=543/64, giving M₄=0.25·(687+647+599+543)/64=594/64≈9.2813, not 619/64. The stated answer is therefore incorrect.qwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-08 — The solution miscomputes the function values at the midpoints and consequently the Riemann sum. The correct values are f(9/8)=687/64, f(11/8)=647/64, f(13/8)=599/64, f(15/8)=543/64, giving M₄=0.25·(687+647+599+543)/64=594/64≈9.2813, not 619/64. The stated answer is therefore incorrect.qwen3.6:27b-mlx: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08gpt-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/riemann_sums, checked 2026-10-08 with SymPy 1.14.0.