Riemann sums: left, right and midpoint
Problem 4.695 · medium
Compute the midpoint Riemann sum \( \displaystyle M_{4} \) for \( \displaystyle f(x) = \frac{5}{x} \) on \( \displaystyle [2, 3] \).
- \[ 1 \cdot \frac{1}{4} = \frac{1}{4} \]Δx = (b − a)/n.✓ Proved
- The midpoint points are x = 17/8, 19/8, 21/8, 23/8.
- \[ \frac{\frac{40}{23} + \frac{40}{21} + \frac{40}{19} + \frac{40}{17}}{4} = \frac{316000}{156009} \]Δx times the sum of the function values there.✓ Proved
- For comparison, the exact integral is -5*log(2) + 5*log(3) ≈ 2.0273.
Answer \( M_{4} = \frac{316000}{156009} \approx 2.0255 \)
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 arithmetic in step 3 is incorrect: the correct midpoint sum gives 315 940/156 009, not 316 000/156 009. The stated value misleads the student about the exact Riemann sum.qwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-08 — The arithmetic in step 3 is incorrect: the correct midpoint sum gives 315 940/156 009, not 316 000/156 009. The stated value misleads the student about the exact Riemann sum.qwen3.6:27b-mlx: pass 2026-10-08gpt-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.
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.