Riemann sums: left, right and midpoint
Problem 4.549 · medium
Compute the midpoint Riemann sum \( \displaystyle M_{6} \) for \( \displaystyle f(x) = 12 - x^{2} \) on \( \displaystyle [0, 2] \).
- \[ 1 \cdot \frac{1}{3} = \frac{1}{3} \]Δx = (b − a)/n.✓ Proved
- The midpoint points are x = 1/6, 1/2, 5/6, 7/6, 3/2, 11/6.Reviewed
- \[ \frac{\frac{311}{36} + \frac{39}{4} + \frac{383}{36} + \frac{407}{36} + \frac{47}{4} + \frac{431}{36}}{3} = \frac{1153}{54} \]Δx times the sum of the function values there.✓ Proved
- For comparison, the exact integral is 64/3 ≈ 21.3333.Reviewed
Answer \( M_{6} = \frac{1153}{54} \approx 21.3519 \)
✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 subinterval width and midpoints, computes the Riemann sum accurately, and provides a valid comparison with the exact integral.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the subinterval width and midpoints, computes the Riemann sum accurately, and provides a valid comparison with the exact integral.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the step size, midpoints, and function evaluations. The final sum and comparison to the exact integral are 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 2026-10-06 with SymPy 1.14.0.