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Riemann sums: left, right and midpoint

Problem 4.849 · medium

Compute the midpoint Riemann sum \( \displaystyle M_{5} \) for \( \displaystyle f(x) = 3 x^{3} + 1 \) on \( \displaystyle [-1, 0] \).

\[ 1 \cdot \frac{1}{5} = \frac{1}{5} \]
Δx = (b − a)/n.✓ Proved
The midpoint points are x = -9/10, -7/10, -1/2, -3/10, -1/10.
Reviewed
\[ \frac{- \frac{1187}{1000} - \frac{29}{1000} + \frac{5}{8} + \frac{919}{1000} + \frac{997}{1000}}{5} = \frac{53}{200} \]
Δx times the sum of the function values there.✓ Proved
For comparison, the exact integral is 1/4 ≈ 0.2500.
Reviewed

Answer \( M_{5} = \frac{53}{200} \approx 0.2650 \)

✓ 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: pass
qwen3.6:27b-mlx: pass

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: pass 2026-10-10
qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the subinterval width, the midpoints, and the resulting Riemann sum. The comparison with the exact integral is accurate.
gpt-oss:20b: pass 2026-10-10

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 2026-10-10 with SymPy 1.14.0.