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Integrals of absolute values

Problem 4.433 · easy

Evaluate \( \displaystyle \int_{2}^{6} \left| x - 3 \right| dx \).

\[ x - 3 \]
The integrand is zero at x = 3.✓ Proved
Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).
\[ \int\limits_{2}^{3} \left(3 - x\right)\, dx + \int\limits_{3}^{6} \left(x - 3\right)\, dx = 5 \]
Integrate each piece and add.✓ Proved

Answer \( 5 \)

Lines: 2 proved, 1 not checked. 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.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	—	a sentence; read, not computed
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 of \|p(x)\|

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution incorrectly refers to x=3 as a 'zero' of the integrand, implying there are multiple zeros or a range between zeros, whereas x=3 is the unique root. This mischaracterization of the function's behavior is mathematically imprecise.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution incorrectly refers to x=3 as a 'zero' of the integrand, implying there are multiple zeros or a range between zeros, whereas x=3 is the unique root. This mischaracterization of the function's behavior is mathematically imprecise.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution incorrectly claims there are 'those zeros' (plural) and implies the sign does not change between them, whereas x-3 has only one zero at x=3. The sign changes from negative to positive at this single point, requiring a split at x=3, not an interval between two zeros.
gpt-oss:20b: pass 2026-10-05

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/absolute_value_integral, checked 2026-10-05 with SymPy 1.14.0.