Riemann sums: left, right and midpoint
Problem 4.403 · medium
Compute the right Riemann sum \( \displaystyle R_{4} \) for \( \displaystyle f(x) = \frac{5}{x} \) on \( \displaystyle [1, 4] \).
- \[ \frac{3}{4} \]Δx = (b − a)/n.✓ Proved
- The right endpoints are x = 7/4, 5/2, 13/4, 4.
- \[ \frac{3 \left(\frac{5}{4} + \frac{20}{13} + 2 + \frac{20}{7}\right)}{4} = \frac{8349}{1456} \]Δx times the sum of the function values there.✓ Proved
- For comparison, the exact integral is 5*log(4) ≈ 6.9315.
Answer \( R_{4} = \frac{8349}{1456} \approx 5.7342 \)
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 is incorrect: the correct value is 8319/1456 ≈ 5.713, not 8349/1456 ≈ 5.734.qwen3.6:27b-mlx: pass — The solution correctly identifies the step size, right endpoints, and function evaluations. The final calculation is correct, and the comparison with the exact integral is accurate.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the step size, right endpoints, and function evaluations. The final calculation is correct, and the comparison with the exact integral is accurate.gpt-oss:20b: fail (error) 2026-10-05 — The computed Riemann sum is incorrect: the correct value is 8319/1456 ≈ 5.713, not 8349/1456 ≈ 5.734.qwen3.6:27b-mlx: fail (error) 2026-10-05 — The step size Δx is calculated as 3/4, but the listed right endpoints (7/4, 5/2, 13/4, 4) correspond to a step size of 1/4. The solution mixes up n=4 (Δx=3/4) with n=12 (Δx=1/4) or simply lists incorrect endpoints for the stated n=4.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.