Integrals of absolute values
Problem 4.751 · easy
Evaluate \( \displaystyle \int_{-2}^{2} \left| 2 x - 2 \right| dx \).
- \[ 2 x - 2 \]The integrand is zero at x = 1.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).
- \[ \int\limits_{1}^{2} \left(2 x - 2\right)\, dx + \int\limits_{-2}^{1} \left(2 - 2 x\right)\, dx = 10 \]Integrate each piece and add.✓ Proved
Answer \( 10 \)
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: fail (misleading) — The sentence claims the sign does not change between zeros, yet the integrand has only one zero at x=1 and changes sign there. This misleads a student into thinking the sign is constant over the entire interval.qwen3.6:27b-mlx: fail (error) — The solution incorrectly states that the integrand has zeros at the bounds or implies multiple zeros, whereas 2x-2 has only one zero at x=1. It also vaguely refers to 'those zeros' (plural) when splitting the integral, which is mathematically imprecise and confusing.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-09 — The sentence claims the sign does not change between zeros, yet the integrand has only one zero at x=1 and changes sign there. This misleads a student into thinking the sign is constant over the entire interval.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution incorrectly states that the integrand has zeros at the bounds or implies multiple zeros, whereas 2x-2 has only one zero at x=1. It also vaguely refers to 'those zeros' (plural) when splitting the integral, which is mathematically imprecise and confusing.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the critical point x=1, splits the integral at this point, and applies the definition of absolute value correctly for each interval. The final calculation is correct.
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.