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Riemann sums: left, right and midpoint

Problem 4.1457 · medium Video

Compute the right Riemann sum \( \displaystyle R_{4} \) for \( \displaystyle f(x) = x^{2} - 3 x + 9 \) on \( \displaystyle [-1, 1] \).

\[ \frac{1 + 1}{4} = \frac{1}{2} \]
Δx = (b − a)/n.✓ Proved
The right endpoints are x = -1/2, 0, 1/2, 1.
\[ \frac{7 + \frac{31}{4} + 9 + \frac{43}{4}}{2} = \frac{69}{4} \]
Δx times the sum of the function values there.✓ Proved
For comparison, the exact integral is 56/3 ≈ 18.6667.

Answer \( R_{4} = \frac{69}{4} \approx 17.2500 \)

The graph of f(x) = x² - 3x + 9 from x = -1 to x = 1. 4 right rectangles approximate the area; their total is 69/4.

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: pass
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-11-07 — reviewer returned a non-object
gpt-oss:20b: pass 2026-11-07
qwen3.6:27b-mlx: inconclusive 2026-11-06 — reviewer returned a non-object
gpt-oss:20b: pass 2026-11-06
qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly identifies the step size, right endpoints, and function values. The arithmetic is correct, and the comparison with the exact integral is accurate.
gpt-oss:20b: pass 2026-11-06

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.