Riemann sums: left, right and midpoint
Problem 4.1005 · medium
Compute the right Riemann sum \( \displaystyle R_{5} \) for \( \displaystyle f(x) = \frac{3}{x} \) on \( \displaystyle [2, 5] \).
- \[ \frac{3}{5} \]Δx = (b − a)/n.✓ Proved
- The right endpoints are x = 13/5, 16/5, 19/5, 22/5, 5.
- \[ \frac{3 \left(\frac{3}{5} + \frac{15}{22} + \frac{15}{19} + \frac{15}{16} + \frac{15}{13}\right)}{5} = \frac{2714373}{1086800} \]Δx times the sum of the function values there.✓ Proved
- For comparison, the exact integral is -3*log(2) + 3*log(5) ≈ 2.7489.
Answer \( R_{5} = \frac{2714373}{1086800} \approx 2.4976 \)
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 computed Riemann sum in step 3 is incorrect; the correct value is approximately 2.6776, not 2.4976.qwen3.6:27b-mlx: pass — The solution correctly identifies the step size, the right endpoints, and computes the Riemann sum accurately. The comparison with the exact integral is also correct.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-13 — The computed Riemann sum in step 3 is incorrect; the correct value is approximately 2.6776, not 2.4976.qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the step size, the right endpoints, and computes the Riemann sum accurately. The comparison with the exact integral is also correct.gpt-oss:20b: fail (error) 2026-10-13 — The arithmetic in step 3 is incorrect; the correct right‑endpoint sum gives R₅≈2.6776, not 2.4976.qwen3.6:27b-mlx: fail (error) 2026-10-13 — The step size is calculated incorrectly as 3/5 instead of 1. Consequently, the right endpoints are wrong (e.g., the first endpoint should be 3, not 13/5), leading to an incorrect Riemann sum.
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 2026-10-13 with SymPy 1.14.0.