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Definite integrals from area formulas

Problem 4.1271 · easy Video

Evaluate \( \displaystyle \int_{-4}^{0} \left|{x + 2}\right| \, dx \) using a formula from geometry.

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

Answer \( 4 \)

The graph of f(x) = Abs(x + 2) from x = -4 to x = 0. Rectangles under the curve get thinner until they fill the area, 4.

✓ 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 reasoning is sound and the final answer is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the geometric shape as two triangles and calculates their areas accurately. The reasoning is sound and the final answer is correct.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the geometric shape as two triangles and calculates their areas accurately. The reasoning is sound and leads to the correct result.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the geometric shape as two triangles and calculates their areas accurately. The reasoning is sound and leads to the correct result.
gpt-oss:20b: pass 2026-10-17

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 2026-10-19 with SymPy 1.14.0.