Riemann sums: left, right and midpoint
Problem 4.1462 · medium Video
Compute the midpoint Riemann sum \( \displaystyle M_{6} \) for \( \displaystyle f(x) = \frac{4}{x} \) on \( \displaystyle [1, 3] \).
- \[ \frac{-1 + 3}{6} = \frac{1}{3} \]Δx = (b − a)/n.✓ Proved
- The midpoint points are x = 7/6, 3/2, 11/6, 13/6, 5/2, 17/6.
- \[ \frac{\frac{24}{17} + \frac{8}{5} + \frac{24}{13} + \frac{24}{11} + \frac{8}{3} + \frac{24}{7}}{3} = \frac{3352768}{765765} \]Δx times the sum of the function values there.✓ Proved
- For comparison, the exact integral is 4*log(3) ≈ 4.3944.
Answer \( M_{6} = \frac{3352768}{765765} \approx 4.3783 \)
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. How we verify
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 sum of function values at the midpoints is incorrect: the first term should be 24/7 (from f(7/6)), not 24/17. This makes the computed Riemann sum wrong.qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-07gpt-oss:20b: fail (error) 2026-11-07 — The sum of function values at the midpoints is incorrect: the first term should be 24/7 (from f(7/6)), not 24/17. This makes the computed Riemann sum wrong.qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly identifies the subinterval width, the midpoints, and the resulting Riemann sum. The comparison with the exact integral is accurate and helpful.gpt-oss:20b: fail (error) 2026-11-06 — The function values are mis‑computed: 4/(7/6)=24/7, not 24/17. The sum in step 3 is therefore incorrect, so the reported Riemann sum is wrong.qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly calculates the step size, identifies the midpoints, and computes the Riemann sum. The comparison with the exact integral is accurate.gpt-oss:20b: fail (error) 2026-11-06 — The function values are mis‑computed: 4/(7/6)=24/7, not 24/17, and the resulting sum times Δx does not give the stated value 3352768/765765. The midpoint sum is therefore incorrect.
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 2026-11-07 with SymPy 1.14.0.