Integrals of absolute values
Problem 4.282 · easy
Evaluate \( \displaystyle \int_{-1}^{3} \left| 4 - 2 x \right| dx \).
- \[ 4 - 2 x \]The integrand is zero at x = 2.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).Reviewed
- \[ \int\limits_{2}^{3} \left(2 x - 4\right)\, dx + \int\limits_{-1}^{2} \left(4 - 2 x\right)\, dx = 10 \]Integrate each piece and add.✓ Proved
Answer \( 10 \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 correct definition of absolute value for each interval. The final calculation is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the root of the linear function, splits the integral at that point, and applies the correct definition of absolute value for each interval. The final calculation is correct.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the root of the linear function, splits the integral at that point, and applies the definition of absolute value to remove the bars. The arithmetic is correct.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.