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Definite integrals from area formulas

Problem 4.1272 · easy Video

Evaluate \( \displaystyle \int_{-4}^{2} \left|{x + 2}\right| \, dx \) 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
\[ \int\limits_{-4}^{2} \left|{x + 2}\right|\, dx = 10 \]
Integrating directly gives the same.✓ Proved

Answer \( 10 \)

The graph of f(x) = Abs(x + 2) from x = -4 to x = 2. 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 geometric interpretation is correct: the integral represents the sum of the areas of two triangles with bases 2 and 4 and heights 2 and 4, respectively. The calculation and conclusion are sound.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The geometric interpretation is correct: the integral represents the sum of the areas of two triangles with bases 2 and 4 and heights 2 and 4, respectively. The calculation and conclusion are sound.
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 reference to '(signed) area' in line 2 is slightly imprecise terminology (since absolute value yields positive area), but the calculation and final result are correct.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution incorrectly claims the signed area is 10. The signed area is actually 0 because the region below the x-axis (from -4 to -2) cancels the region above (from -2 to 2). The problem asks for the integral of the absolute value, which corresponds to the total geometric area, but the text explicitly mislabels the calculation as '(signed) area'.
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.