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Definite integrals from area formulas

Problem 4.1069 · easy Video

Evaluate \( \displaystyle \int_{0}^{6} \left|{x - 2}\right| \, dx \) using a formula from geometry.

The region between the graph and the x-axis is two triangles meeting at x = 2.
Reviewed
\[ \frac{2^{2}}{2} + \frac{4^{2}}{2} = 10 \]
Its (signed) area.✓ Proved
\[ \int\limits_{0}^{6} \left|{x - 2}\right|\, dx = 10 \]
Integrating directly gives the same.✓ Proved

Answer \( 10 \)

The graph of f(x) = Abs(x - 2) from x = 0 to x = 6. Rectangles under the curve get thinner until they fill the area, 10.

Lines: 2 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the geometric shapes (two triangles) and calculates their areas accurately to find the total area under the absolute value curve.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the geometric shapes (two triangles) and calculates their areas accurately to find the total area under the absolute value curve.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the geometric shapes (two triangles) and calculates their areas accurately to evaluate the integral.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (misleading) 2026-10-14 — The solution claims the area is '(signed) area', but the integral of an absolute value is always non-negative, so 'signed' is incorrect terminology here. Furthermore, the geometric setup describes two triangles, but the calculation `2**2/2 + 4**2/2` implicitly assumes the heights are 2 and 4 without explicitly stating that the height at x=0 is |-2|=2 and at x=6 is |4|=4, which is a minor gap in rigor but the main issue is the misleading 'signed' label for a strictly positive quantity.
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.