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Definite integrals from area formulas

Problem 4.1192 · 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
\[ \frac{1^{2}}{2} + \frac{2^{2}}{2} = \frac{5}{2} \]
Its (signed) area.✓ Proved
\[ \int\limits_{-4}^{-1} \left|{x + 2}\right|\, dx = \frac{5}{2} \]
Integrating directly gives the same.✓ Proved

Answer \( \frac{5}{2} \)

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, 5/2.

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 shape as two triangles and calculates their areas accurately. The setup and conclusion are sound.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the geometric shape as two triangles and calculates their areas accurately. The setup and conclusion are sound.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the geometric shape as two triangles with bases and heights derived from the vertex at x=-2 and the integration bounds. The calculation of the areas (1/2 * 2 * 2 + 1/2 * 1 * 1 = 2.5) is correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the geometric shape as two triangles with bases and heights derived from the vertex at x=-2 and the integration bounds. The calculation of the areas (1/2 * 2 * 2 + 1/2 * 1 * 1 = 2.5) is correct.
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.