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Definite integrals from area formulas

Problem 4.1067 · easy Video

Evaluate \( \displaystyle \int_{-4}^{1} \left|{x + 2}\right| \, dx \) using a formula from geometry.

The region between the graph and the x-axis is two triangles meeting at x = -2.
Reviewed
\[ \frac{2^{2}}{2} + \frac{3^{2}}{2} = \frac{13}{2} \]
Its (signed) area.✓ Proved
\[ \int\limits_{-4}^{1} \left|{x + 2}\right|\, dx = \frac{13}{2} \]
Integrating directly gives the same.✓ Proved

Answer \( \frac{13}{2} \)

The graph of f(x) = Abs(x + 2) from x = -4 to x = 1. Rectangles under the curve get thinner until they fill the area, 13/2.

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 their dimensions based on the roots and bounds. The calculation of the areas is correct, and the final result matches the stated answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the geometric shapes (two triangles) and their dimensions based on the roots and bounds. The calculation of the areas is correct, and the final result matches the stated answer.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (misleading) 2026-10-14 — The solution claims the result is the '(signed) area', but the integral of an absolute value is always non-negative and represents geometric area, not signed area. Signed area would imply negative contributions, which contradicts the use of absolute value.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution claims the area is '(signed) area', but the integral of an absolute value represents total geometric area, not signed area. While the numerical calculation is correct, the terminology is mathematically incorrect and misleading.
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.