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Definite integrals from area formulas

Problem 4.1185 · easy Video

Evaluate \( \displaystyle \int_{1}^{5} - 2 x \, dx \) using a formula from geometry.

The region between the graph and the x-axis is a trapezoid, below the axis.
\[ \frac{\left(-10 - 2\right) \left(-1 + 5\right)}{2} = -24 \]
Its (signed) area.✓ Proved
\[ \int\limits_{1}^{5} \left(- 2 x\right)\, dx = -24 \]
Integrating directly gives the same.✓ Proved

Answer \( -24 \)

The graph of f(x) = -2x from x = 1 to x = 5. Rectangles under the curve get thinner until they fill the area, -24. 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: inconclusive — reviewer returned a non-object

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the region as a trapezoid and applies the geometric area formula with the correct sign for the region below the x-axis.
gpt-oss:20b: fail (error) 2026-10-16 — The description of the region as a trapezoid is incorrect; the shape bounded by the line, the x‑axis, and the vertical lines x=1 and x=5 is a general quadrilateral, not a trapezoid. The area formula used assumes parallel bases, which does not hold here.
qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the region as a single trapezoid. The function f(x) = -2x is linear, so the region between the graph and the x-a
gpt-oss:20b: pass 2026-10-15

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.