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Riemann sums: left, right and midpoint

Problem 4.1458 · medium Video

Compute the midpoint Riemann sum \( \displaystyle M_{4} \) for \( \displaystyle f(x) = - 3 x^{2} - 3 x + 11 \) on \( \displaystyle [0, 2] \).

\[ \frac{0 + 2}{4} = \frac{1}{2} \]
Δx = (b − a)/n.✓ Proved
The midpoint points are x = 1/4, 3/4, 5/4, 7/4.
Reviewed
\[ \frac{- \frac{55}{16} + \frac{41}{16} + \frac{113}{16} + \frac{161}{16}}{2} = \frac{65}{8} \]
Δx times the sum of the function values there.✓ Proved
For comparison, the exact integral is 8 ≈ 8.0000.
Reviewed

Answer \( M_{4} = \frac{65}{8} \approx 8.1250 \)

The graph of f(x) = -3x² - 3x + 11 from x = 0 to x = 2. 4 midpoint rectangles approximate the area; their total is 65/8.

✓ 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 calculates the step size, identifies the midpoints, computes the Riemann sum, and provides the exact integral for comparison. All steps are mathematically sound.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly calculates the step size, identifies the midpoints, computes the Riemann sum, and provides the exact integral for comparison. All steps are mathematically sound.
gpt-oss:20b: pass 2026-11-07
qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly identifies the subinterval width and midpoints, computes the Riemann sum accurately, and provides a correct comparison with the exact integral.
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.