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Riemann sums: left, right and midpoint

Problem 4.1461 · medium Video

Compute the right Riemann sum \( \displaystyle R_{6} \) for \( \displaystyle f(x) = x^{3} + 1 \) on \( \displaystyle [-1, 1] \).

\[ \frac{1 + 1}{6} = \frac{1}{3} \]
Δx = (b − a)/n.✓ Proved
The right endpoints are x = -2/3, -1/3, 0, 1/3, 2/3, 1.
\[ \frac{\frac{19}{27} + \frac{26}{27} + 1 + \frac{28}{27} + \frac{35}{27} + 2}{3} = \frac{7}{3} \]
Δx times the sum of the function values there.✓ Proved
For comparison, the exact integral is 2 ≈ 2.0000.

Answer \( R_{6} = \frac{7}{3} \approx 2.3333 \)

The graph of f(x) = x³ + 1 from x = -1 to x = 1. 6 right rectangles approximate the area; their total is 7/3.

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: fail (error) — The computed Riemann sum is incorrect: the sum of f-values is 199/27, so R₆ = (1/3)(199/27)=199/81 ≈ 2.4568, not 7/3. The setup of Δx and right endpoints is correct, but the arithmetic in step 3 is wrong.
qwen3.6:27b-mlx: pass — The solution correctly identifies the subinterval width, the right endpoints, and computes the sum accurately. The comparison with the exact integral is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly identifies the subinterval width, the right endpoints, and computes the sum accurately. The comparison with the exact integral is correct.
gpt-oss:20b: fail (error) 2026-11-07 — The computed Riemann sum is incorrect: the sum of f-values is 199/27, so R₆ = (1/3)(199/27)=199/81 ≈ 2.4568, not 7/3. The setup of Δx and right endpoints is correct, but the arithmetic in step 3 is wrong.
qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly identifies the step size, the right endpoints, and the resulting sum. The comparison with the exact integral is accurate.
gpt-oss:20b: pass 2026-11-06
qwen3.6:27b-mlx: inconclusive 2026-11-06 — reviewer returned a non-object
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.