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Riemann sums: left, right and midpoint

Problem 4.1463 · medium Video

Compute the midpoint Riemann sum \( \displaystyle M_{6} \) for \( \displaystyle f(x) = - 2 x^{2} + x + 9 \) on \( \displaystyle [-1, 1] \).

\[ \frac{1 + 1}{6} = \frac{1}{3} \]
Δx = (b − a)/n.✓ Proved
The midpoint points are x = -5/6, -1/2, -1/6, 1/6, 1/2, 5/6.
\[ \frac{\frac{61}{9} + 8 + \frac{76}{9} + \frac{79}{9} + 9 + \frac{82}{9}}{3} = \frac{451}{27} \]
Δx times the sum of the function values there.✓ Proved
For comparison, the exact integral is 50/3 ≈ 16.6667.

Answer \( M_{6} = \frac{451}{27} \approx 16.7037 \)

The graph of f(x) = -2x² + x + 9 from x = -1 to x = 1. 6 midpoint rectangles approximate the area; their total is 451/27.

Lines: 2 proved, 2 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	—	a sentence; read, not computed
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	Not checked	—	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: inconclusive — reviewer returned a non-object
qwen3.6:27b-mlx: pass — The solution correctly calculates the step size, identifies the correct midpoints for the given interval and number of subintervals, and computes the Riemann sum accurately. The comparison with the exact integral is also correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly calculates the step size, identifies the correct midpoints for the given interval and number of subintervals, and computes the Riemann sum accurately. The comparison with the exact integral is also correct.
gpt-oss:20b: inconclusive 2026-11-07 — reviewer returned a non-object
qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly identifies the subinterval width, the midpoints, and the resulting Riemann sum. The comparison with the exact integral is accurate and helpful context.
gpt-oss:20b: pass 2026-11-06
qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly calculates the subinterval width, identifies the correct midpoints for the given interval and partition size, and computes the Riemann sum accurately. The comparison with the exact integral is also correct.
gpt-oss:20b: fail (error) 2026-11-06 — The computed midpoint sum is incorrect: the correct value is 442/27, not 451/27. The solution’s arithmetic in step 3 is flawed.

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 2026-11-07 with SymPy 1.14.0.