Limits of Riemann sums as integrals
Problem 4.1061 · medium
Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{\left(-1 + \frac{i - 1}{n}\right)^{2}}{n} \) as a definite integral, and evaluate it.
- Δx = 1/n, and x_i = -1 + 1(i − 1)/n runs across [-1, 0] (left endpoints). The sum is Σ f(x_i)Δx with f(x) = x**2.
- \[ \int\limits_{-1}^{0} x^{2}\, dx = \frac{1}{3} \]Evaluate the integral.✓ Proved
- \[ \lim_{n \to \infty} \sum_{i=1}^{n} \frac{\left(-1 + \frac{i - 1}{n}\right)^{2}}{n} = \frac{1}{3} \]The limit of the sum itself, from the summation formulas, agrees.✓ Proved
Answer \( \int_{-1}^{0} x^{2}\, dx = \frac{1}{3} \)
Lines: 2 proved, 1 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the Riemann sum itself with n = 200,000, within its error bound |
Reviewers
gpt-oss:20b: inconclusive — reviewer returned a non-objectqwen3.6:27b-mlx: pass — The solution correctly identifies the Riemann sum as corresponding to the integral of x^2 from -1 to 0 using left endpoints. The evaluation is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the Riemann sum as corresponding to the integral of x^2 from -1 to 0 using left endpoints. The evaluation is correct.gpt-oss:20b: inconclusive 2026-10-17 — reviewer returned a non-objectqwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the Riemann sum structure, the function, the interval, and the evaluation. The setup and conclusion are sound.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the Riemann sum as corresponding to the integral of x^2 from -1 to 0 using left endpoints. The evaluation is correct.gpt-oss:20b: pass 2026-10-14
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_limit_integral, checked 2026-10-17 with SymPy 1.14.0.