Riemann sums: left, right and midpoint
Problem 4.1010 · medium
Compute the right Riemann sum \( \displaystyle R_{4} \) for \( \displaystyle f(x) = 12 - x^{2} \) on \( \displaystyle [0, 3] \).
- \[ \frac{3}{4} \]Δx = (b − a)/n.✓ Proved
- The right endpoints are x = 3/4, 3/2, 9/4, 3.Reviewed
- \[ \frac{3 \left(3 + \frac{111}{16} + \frac{39}{4} + \frac{183}{16}\right)}{4} = \frac{747}{32} \]Δx times the sum of the function values there.✓ Proved
- For comparison, the exact integral is 27 ≈ 27.0000.Reviewed
Answer \( R_{4} = \frac{747}{32} \approx 23.3438 \)
Lines: 2 proved, 2 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the step size, right endpoints, and function values. The arithmetic is correct, and the comparison with the exact integral is accurate.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the step size, right endpoints, and function values. The arithmetic is correct, and the comparison with the exact integral is accurate.gpt-oss:20b: fail (error) 2026-10-13 — The computed Riemann sum in step 3 is incorrect: the sum of the function values is 231/2, and multiplying by Δx=3/4 gives 693/8 (≈86.625), not 747/32. The solution’s arithmetic is wrong.qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the step size, right endpoints, and function values. The final calculation is correct, and 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 2026-10-13 with SymPy 1.14.0.