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Definite integrals from area formulas

Problem 4.1372 · easy Video

Evaluate \( \displaystyle \int_{0}^{4} 6 - x \, dx \) using a formula from geometry.

The region between the graph and the x-axis is a trapezoid.
\[ \frac{\left(0 + 4\right) \left(2 + 6\right)}{2} = 16 \]
Its (signed) area.✓ Proved
\[ \int\limits_{0}^{4} \left(6 - x\right)\, dx = 16 \]
Integrating directly gives the same.✓ Proved

Answer \( 16 \)

The graph of f(x) = 6 - x from x = 0 to x = 4. Rectangles under the curve get thinner until they fill the area, 16.

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 response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The setup incorrectly identifies the region as a trapezoid; the region under y=6-x from x=0 to x=4 is a trapezoid with heights 6 and 2, but the calc

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-29 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The setup incorrectly identifies the region as a trapezoid; the region under y=6-x from x=0 to x=4 is a trapezoid with heights 6 and 2, but the calc
gpt-oss:20b: pass 2026-10-29
qwen3.6:27b-mlx: inconclusive 2026-10-28 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-28
qwen3.6:27b-mlx: inconclusive 2026-10-28 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-28

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 2026-10-29 with SymPy 1.14.0.