Integrals of absolute values
Problem 4.590 · easy
Evaluate \( \displaystyle \int_{-4}^{-1} \left| - 2 x - 6 \right| dx \).
- \[ - 2 x - 6 \]The integrand is zero at x = -3.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).Reviewed
- \[ \int\limits_{-4}^{-3} \left(- 2 x - 6\right)\, dx + \int\limits_{-3}^{-1} \left(2 x + 6\right)\, dx = 5 \]Integrate each piece and add.✓ Proved
Answer \( 5 \)
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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the root of the linear function, splits the integral at that point, and applies the absolute value definition correctly for each sub-interval. The final calculation is accurate.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the root of the linear function, splits the integral at that point, and applies the absolute value definition correctly for each sub-interval. The final calculation is accurate.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution incorrectly states that the integrand is zero at x = -3, implying this is the only zero, but fails to acknowledge that the sign of the linear function changes at this point. More critically, the sentence claims 'Between those zeros the sign does not change', which is nonsensical as there is only one zero mentioned. The correct logic is that the sign changes at x = -3. The calculation splits the integral at x = -3, which is correct, but the justification provided in step 2 is mathematically incoherent and misleading regarding the behavior of the absolute value function.
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 2026-10-07 with SymPy 1.14.0.