Integrals of absolute values
Problem 4.290 · easy
Evaluate \( \displaystyle \int_{2}^{4} \left| x - 3 \right| dx \).
- \[ x - 3 \]The integrand is zero at x = 3.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).
- \[ \int\limits_{3}^{4} \left(x - 3\right)\, dx + \int\limits_{2}^{3} \left(3 - x\right)\, dx = 1 \]Integrate each piece and add.✓ Proved
Answer \( 1 \)
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 refers to x=3 as a 'zero' of the integrand, implying the function is zero in a neighborhood, which is false. The absolute value function is not differentiable at x=3, so the sign change logic is flawed; it should simply split the integral at the point where the argument of the absolute value is zero.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution incorrectly refers to x=3 as a 'zero' of the integrand, implying the function is zero in a neighborhood, which is false. The absolute value function is not differentiable at x=3, so the sign change logic is flawed; it should simply split the integral at the point where the argument of the absolute value is zero.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution incorrectly refers to x=3 as a 'zero' of the integrand, implying the function is zero over an interval or that the sign change is between multiple zeros. The absolute value function |x-3| is zero only at the single point x=3, and the sign changes at this point, not 'between zeros'. This is a fundamental conceptual error in describing the behavior of the integrand.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.