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Riemann sums: left, right and midpoint

Problem 4.1460 · medium Video

Compute the right Riemann sum \( \displaystyle R_{6} \) for \( \displaystyle f(x) = \frac{2}{x} \) on \( \displaystyle [1, 2] \).

\[ \frac{-1 + 2}{6} = \frac{1}{6} \]
Δx = (b − a)/n.✓ Proved
The right endpoints are x = 7/6, 4/3, 3/2, 5/3, 11/6, 2.
Reviewed
\[ \frac{1 + \frac{12}{11} + \frac{6}{5} + \frac{4}{3} + \frac{3}{2} + \frac{12}{7}}{6} = \frac{18107}{13860} \]
Δx times the sum of the function values there.✓ Proved
For comparison, the exact integral is 2*log(2) ≈ 1.3863.
Reviewed

Answer \( R_{6} = \frac{18107}{13860} \approx 1.3064 \)

The graph of f(x) = 2/x from x = 1 to x = 2. 6 right rectangles approximate the area; their total is 18107/13860.

✓ 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. How we verify

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 — The solution correctly identifies the step size, the right endpoints, and the resulting sum. The comparison with the exact integral is accurate and helpful.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly identifies the step size, the right endpoints, and the resulting sum. The comparison with the exact integral is accurate and helpful.
gpt-oss:20b: pass 2026-11-07
qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly identifies the step size, the right endpoints, and the corresponding function values. The final calculation and comparison with the exact integral are accurate.
gpt-oss:20b: pass 2026-11-06
qwen3.6:27b-mlx: pass 2026-11-06
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.