Riemann sums: left, right and midpoint
Problem 4.541 · medium
Compute the left Riemann sum \( \displaystyle L_{4} \) for \( \displaystyle f(x) = 3 x^{2} - 2 x + 13 \) on \( \displaystyle [-1, 1] \).
- \[ 1 \cdot \frac{1}{2} = \frac{1}{2} \]Δx = (b − a)/n.✓ Proved
- The left endpoints are x = -1, -1/2, 0, 1/2.Reviewed
- \[ \frac{\frac{51}{4} + 13 + \frac{59}{4} + 18}{2} = \frac{117}{4} \]Δx times the sum of the function values there.✓ Proved
- For comparison, the exact integral is 28 ≈ 28.0000.Reviewed
Answer \( L_{4} = \frac{117}{4} \approx 29.2500 \)
✓ 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 step size, left endpoints, and function values. The final comparison with the exact integral is mathematically correct and serves as a valid sanity check.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the step size, left endpoints, and function values. The final comparison with the exact integral is mathematically correct and serves as a valid sanity check.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the step size, left 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 2026-10-06 with SymPy 1.14.0.