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Definite integrals from area formulas

Problem 4.1194 · easy Video

Evaluate \( \displaystyle \int_{-3}^{2} \left|{x}\right| \, dx \) using a formula from geometry.

The region between the graph and the x-axis is two triangles meeting at x = 0.
Reviewed
\[ \frac{2^{2}}{2} + \frac{3^{2}}{2} = \frac{13}{2} \]
Its (signed) area.✓ Proved
\[ \int\limits_{-3}^{2} \left|{x}\right|\, dx = \frac{13}{2} \]
Integrating directly gives the same.✓ Proved

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

The graph of f(x) = Abs(x) from x = -3 to x = 2. 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 shape as two triangles and calculates their areas. The phrasing 'signed area' in line 2 is slightly imprecise since absolute value yields positive area, but the calculation is correct and the final result matches the problem statement.

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. The phrasing 'signed area' in line 2 is slightly imprecise since absolute value yields positive area, but the calculation is correct and the final result matches the problem statement.
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 and calculates their areas accurately. The reference to '(signed) area' in line 2 is slightly imprecise terminology for the sum of absolute areas, but the calculation and final result are correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (misleading) 2026-10-15 — The solution claims the result is the '(signed) area', but the integral of |x| is strictly positive, whereas signed area can be negative. This conflates two distinct concepts and teaches incorrect terminology.
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.