Home›Calculus 1›Limits of Riemann sums as integrals›Problem 4.1289

Limits of Riemann sums as integrals

Problem 4.1289 · medium

Write \( \displaystyle \lim_{n\to\infty} \sum_{i=1}^{n} \frac{9 \left(1 + \frac{3 i - 3}{n}\right)^{2} - 3}{n} \) as a definite integral, and evaluate it.

Δx = 3/n, and x_i = 1 + 3(i − 1)/n runs across [1, 4] (left endpoints). The sum is Σ f(x_i)Δx with f(x) = 3*x**2 - 1.
\[ \int\limits_{1}^{4} \left(3 x^{2} - 1\right)\, dx = 60 \]
Evaluate the integral.✓ Proved
\[ \lim_{n \to \infty} \sum_{i=1}^{n} \frac{9 \left(1 + \frac{3 i - 3}{n}\right)^{2} - 3}{n} = 60 \]
The limit of the sum itself, from the summation formulas, agrees.✓ Proved

Answer \( \int_{1}^{4} 3 x^{2} - 1\, dx = 60 \)

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: fail (error) — The setup incorrectly identifies Δx and the integrand. The sum has a factor 1/n, not 3/n, and the integrand should be 9x²‑3, not 3x²‑1. This misidentification leads to an incorrect integral representation.
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-20 — reviewer returned a non-object
gpt-oss:20b: fail (error) 2026-10-20 — The setup incorrectly identifies Δx and the integrand. The sum has a factor 1/n, not 3/n, and the integrand should be 9x²‑3, not 3x²‑1. This misidentification leads to an incorrect integral representation.
qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-19

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 2026-10-20 with SymPy 1.14.0.