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Riemann sums: left, right and midpoint

Problem 4.1456 · medium Video

Compute the left Riemann sum \( \displaystyle L_{4} \) for \( \displaystyle f(x) = - 2 x^{2} + 3 x + 12 \) on \( \displaystyle [1, 3] \).

\[ \frac{-1 + 3}{4} = \frac{1}{2} \]
Δx = (b − a)/n.✓ Proved
The left endpoints are x = 1, 3/2, 2, 5/2.
\[ \frac{7 + 10 + 12 + 13}{2} = 21 \]
Δx times the sum of the function values there.✓ Proved
For comparison, the exact integral is 56/3 ≈ 18.6667.

Answer \( L_{4} = 21 \approx 21.0000 \)

The graph of f(x) = -2x² + 3x + 12 from x = 1 to x = 3. 4 left rectangles approximate the area; their total is 21.

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: inconclusive — reviewer returned a non-object
qwen3.6:27b-mlx: pass — The solution correctly identifies the step size, left endpoints, and function values. The calculation of the Riemann sum is correct, and the comparison with the exact integral is accurate.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-07 — The solution correctly identifies the step size, left endpoints, and function values. The calculation of the Riemann sum is correct, and the comparison with the exact integral is accurate.
gpt-oss:20b: inconclusive 2026-11-07 — reviewer returned a non-object
qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly identifies the step size, left endpoints, and function values. The calculation of the Riemann sum is accurate, and the comparison with the exact integral is correct.
gpt-oss:20b: inconclusive 2026-11-06 — reviewer returned a non-object
qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly identifies the step size, left endpoints, and function values. The final sum is calculated correctly.
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.