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Riemann sums: left, right and midpoint

Problem 4.401 · medium

Compute the left Riemann sum \( \displaystyle L_{4} \) for \( \displaystyle f(x) = 12 - x^{2} \) on \( \displaystyle [0, 1] \).

\[ 1 \cdot \frac{1}{4} = \frac{1}{4} \]
Δx = (b − a)/n.✓ Proved
The left endpoints are x = 0, 1/4, 1/2, 3/4.
Reviewed
\[ \frac{\frac{183}{16} + \frac{47}{4} + \frac{191}{16} + 12}{4} = \frac{377}{32} \]
Δx times the sum of the function values there.✓ Proved
For comparison, the exact integral is 35/3 ≈ 11.6667.
Reviewed

Answer \( L_{4} = \frac{377}{32} \approx 11.7812 \)

✓ 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.

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, left endpoints, and function values. The final calculation is correct, and the comparison with the exact integral is accurate.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the step size, left endpoints, and function values. The final calculation is correct, and the comparison with the exact integral is accurate.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.