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Definite integrals from area formulas

Problem 4.1146 · easy Video

Evaluate \( \displaystyle \int_{4}^{5} 1 - x \, dx \) using a formula from geometry.

The region between the graph and the x-axis is a trapezoid, below the axis.
\[ \frac{\left(-4 + 5\right) \left(-4 - 3\right)}{2} = - \frac{7}{2} \]
Its (signed) area.✓ Proved
\[ \int\limits_{4}^{5} \left(1 - x\right)\, dx = - \frac{7}{2} \]
Integrating directly gives the same.✓ Proved

Answer \( - \frac{7}{2} \)

The graph of f(x) = 1 - x from x = 4 to x = 5. Rectangles under the curve get thinner until they fill the area, -7/2. Area below the x-axis counts as negative.

Lines: 2 proved, 1 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	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	numerical quadrature

Reviewers

gpt-oss:20b: fail (misleading) — The description of the region as a trapezoid with horizontal bases is inaccurate; the segment of the line y=1−x between x=4 and 5 is not horizontal, so the area formula used is not justified. The integral result is correct, but the geometric reasoning is misleading.
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the region as a trapezoid; the area under a linear function is a trapezoid, but the calculation in line 2 uses t

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the region as a trapezoid; the area under a linear function is a trapezoid, but the calculation in line 2 uses t
gpt-oss:20b: fail (misleading) 2026-10-18 — The description of the region as a trapezoid with horizontal bases is inaccurate; the segment of the line y=1−x between x=4 and 5 is not horizontal, so the area formula used is not justified. The integral result is correct, but the geometric reasoning is misleading.
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-14

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/integral_geometry, checked 2026-10-18 with SymPy 1.14.0.