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Definite integrals from area formulas

Problem 4.1065 · easy Video

Evaluate \( \displaystyle \int_{1}^{5} - 3 x \, dx \) using a formula from geometry.

The region between the graph and the x-axis is a trapezoid, below the axis.
\[ \frac{\left(-15 - 3\right) \left(-1 + 5\right)}{2} = -36 \]
Its (signed) area.✓ Proved
\[ \int\limits_{1}^{5} \left(- 3 x\right)\, dx = -36 \]
Integrating directly gives the same.✓ Proved

Answer \( -36 \)

The graph of f(x) = -3x from x = 1 to x = 5. Rectangles under the curve get thinner until they fill the area, -36. 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: pass
qwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the region as a trapezoid; the graph of y = -3x is a line, so the region bounded by the x-axis, x=1, and x=5 is a triangle (with vertices at (1,0), (5,0), and (1,-3)). While the numerical result is correct, the geometric classification is wrong.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution incorrectly identifies the region as a trapezoid; the graph of y = -3x is a line, so the region bounded by the x-axis, x=1, and x=5 is a triangle (with vertices at (1,0), (5,0), and (1,-3)). While the numerical result is correct, the geometric classification is wrong.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims the region is a trapezoid, but the graph of y = -3x is a line, so the region between the graph and the x-axis from x=1 to x=5 is
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the region as a trapezoid. The function f(x) = -3x is linear, so the region bounded by the graph, the x-axis, an
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 2026-10-17 with SymPy 1.14.0.