Riemann sums: left, right and midpoint
Problem 4.547 · medium
Compute the left Riemann sum \( \displaystyle L_{5} \) for \( \displaystyle f(x) = - x^{2} + x + 13 \) on \( \displaystyle [0, 2] \).
- \[ \frac{2}{5} \]Δx = (b − a)/n.✓ Proved
- The left endpoints are x = 0, 2/5, 4/5, 6/5, 8/5.Reviewed
- \[ \frac{2 \left(\frac{301}{25} + \frac{319}{25} + 13 + \frac{329}{25} + \frac{331}{25}\right)}{5} = \frac{642}{25} \]Δx times the sum of the function values there.✓ Proved
- For comparison, the exact integral is 76/3 ≈ 25.3333.Reviewed
Answer \( L_{5} = \frac{642}{25} \approx 25.6800 \)
✓ 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
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06gpt-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, leading to the correct Riemann sum. The comparison with the exact integral is accurate and helpful context.
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.