Integrals of absolute values
Problem 4.438 · easy
Evaluate \( \displaystyle \int_{1}^{6} \left| 3 - x \right| dx \).
- \[ 3 - x \]The integrand is zero at x = 3.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).
- \[ \int\limits_{1}^{3} \left(3 - x\right)\, dx + \int\limits_{3}^{6} \left(x - 3\right)\, dx = \frac{13}{2} \]Integrate each piece and add.✓ Proved
Answer \( \frac{13}{2} \)
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: passqwen3.6:27b-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 them. The absolute value function |3-x| has only one zero at x=3, so the phrase 'Between those zeros' is mathematically nonsensical and misleading.
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 them. The absolute value function |3-x| has only one zero at x=3, so the phrase 'Between those zeros' is mathematically nonsensical and misleading.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution incorrectly claims the integrand is zero at x=3 and implies there are multiple zeros or a range 'between' them, whereas 3-x has only one zero at x=3. The logic for splitting the integral is correct, but the justification is mathematically incoherent.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.