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Riemann sums: left, right and midpoint

Problem 4.1454 · medium Video

Compute the right Riemann sum \( \displaystyle R_{5} \) for \( \displaystyle f(x) = 12 - x^{2} \) on \( \displaystyle [1, 4] \).

\[ \frac{-1 + 4}{5} = \frac{3}{5} \]
Δx = (b − a)/n.✓ Proved
The right endpoints are x = 8/5, 11/5, 14/5, 17/5, 4.
Reviewed
\[ \frac{3 \left(-4 + \frac{11}{25} + \frac{104}{25} + \frac{179}{25} + \frac{236}{25}\right)}{5} = \frac{258}{25} \]
Δx times the sum of the function values there.✓ Proved
For comparison, the exact integral is 15 ≈ 15.0000.
Reviewed

Answer \( R_{5} = \frac{258}{25} \approx 10.3200 \)

The graph of f(x) = 12 - x² from x = 1 to x = 4. 5 right rectangles approximate the area; their total is 258/25.

Lines: 2 proved, 2 reviewed. 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	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 computes the Riemann sum accurately. The comparison with the exact integral is correct 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 computes the Riemann sum accurately. The comparison with the exact integral is correct 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 computes the Riemann sum accurately. The comparison with the exact integral is also correct.
gpt-oss:20b: fail (error) 2026-11-06 — The computed sum of function values is incorrect. 12‑x² evaluated at the right endpoints gives 236/25, 179/25, 104/25, 11/25, and –4, whose sum is 526/25, not the 258/25 used. Consequently the Riemann sum should be (3/5)(526/25)=1578/125≈12.624, not 10.32.
qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly identifies the step size and right endpoints, computes the Riemann sum accurately, and provides a valid comparison with the exact integral.
gpt-oss:20b: fail (error) 2026-11-06 — The Riemann sum is computed incorrectly: the function values at the right endpoints are mis‑summed, leading to an incorrect final value. The correct sum gives R₅=12.624, not 10.32.

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.