Integrals of absolute values
Problem 4.744 · easy
Evaluate \( \displaystyle \int_{-4}^{1} \left| - 2 x - 2 \right| dx \).
- \[ - 2 x - 2 \]The integrand is zero at x = -1.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).Reviewed
- \[ \int\limits_{-1}^{1} \left(2 x + 2\right)\, dx + \int\limits_{-4}^{-1} \left(- 2 x - 2\right)\, dx = 13 \]Integrate each piece and add.✓ Proved
Answer \( 13 \)
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 integrand, 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 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the root of the integrand, 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 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — The sentence claims the sign does not change 'between those zeros,' but a linear function has only one zero. The sign changes at x = -1, so the absolute value must be split into two cases: -( -2x - 2 ) for x < -1 and ( -2x - 2 ) for x > -1. The solution incorrectly implies the sign is constant across the entire interval or misidentifies the splitting logic.
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 2026-10-09 with SymPy 1.14.0.