Integrals of absolute values
Problem 4.434 · easy
Evaluate \( \displaystyle \int_{-3}^{0} \left| - x - 1 \right| dx \).
- \[ - x - 1 \]The integrand is zero at x = -1.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).
- \[ \int\limits_{-1}^{0} \left(x + 1\right)\, dx + \int\limits_{-3}^{-1} \left(- x - 1\right)\, dx = \frac{5}{2} \]Integrate each piece and add.✓ Proved
Answer \( \frac{5}{2} \)
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 claims the integrand is zero at x = -1 and implies there are multiple zeros, but -x - 1 = 0 only at x = -1. Furthermore, the sign analysis is flawed: for x in [-3, -1), -x - 1 > 0, so | -x - 1 | = -x - 1, but the solution integrates -x - 1 on [-3, -1] (correct) and x + 1 on [-1, 0] (correct for |x+1|), yet the text 'Between those zeros' is nonsensical as there is only one zero. The primary error is the confusing and incorrect description of the roots and sign changes.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution incorrectly claims the integrand is zero at x = -1 and implies there are multiple zeros, but -x - 1 = 0 only at x = -1. Furthermore, the sign analysis is flawed: for x in [-3, -1), -x - 1 > 0, so | -x - 1 | = -x - 1, but the solution integrates -x - 1 on [-3, -1] (correct) and x + 1 on [-1, 0] (correct for |x+1|), yet the text 'Between those zeros' is nonsensical as there is only one zero. The primary error is the confusing and incorrect description of the roots and sign changes.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-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 there is only one zero at x = -1 where the sign changes from positive to negative. This mischaracterization of the absolute value function's behavior leads to a confusing explanation, even though the final split of the integral is correct.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.