Integrals of absolute values
Problem 4.283 · easy
Evaluate \( \displaystyle \int_{-3}^{-1} \left| - 2 x - 4 \right| dx \).
- \[ - 2 x - 4 \]The integrand is zero at x = -2.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).
- \[ \int\limits_{-3}^{-2} \left(- 2 x - 4\right)\, dx + \int\limits_{-2}^{-1} \left(2 x + 4\right)\, dx = 2 \]Integrate each piece and add.✓ Proved
Answer \( 2 \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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: inconclusive — reviewer returned a non-objectqwen3.6:27b-mlx: inconclusive — reviewer returned a non-object
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer returned a non-objectgpt-oss:20b: inconclusive 2026-10-04 — reviewer returned a non-objectqwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution incorrectly states that the integrand is zero at x = -2 and implies this is the only zero, but fails to explicitly split the integral at this point in the setup description, although the calculation does split it. More critically, the sentence 'Between those zeros' is confusing because there is only one zero mentioned. The sign analysis is correct, but the phrasing is imprecise. However, the main issue is that the sentence claims 'Between those zeros' implying multiple zeros, but only one is listed. Actually, looking closer, the sentence says 'Between those zeros' but only identifies x=-2. This is a minor style issue. Let's look at the math. The integral of |-2x-4| from -3 to -1. Zero at -2. From -3 to -2, -2x-4 is positive (e.g. x=-3 => 6-4=2). So |-2x-4| = -2x-4. From -2 to -1, -2x-4 is negative (e.g. x=-1 => 2-4=-2). So |-2x-4| = -(-2x-4) = 2x+4. The solution sets up Integral(-2*x - 4, (x, -3, -2)) + Integral(2*x + 4, (x, -2, -1)). This is correct. The result is 2. The sentence 'Between those zeros' is weird because there is only one zero. It should say 'At the zero x=-2, the sign changes'. The sentence is misleading/confusing. Is it an error? The math is right. The text is sloppy. 'Between those zeros' suggests there are at least two zeros. There is only one. This is a style/clarity issue, but could be misleading. Let's call it misleading.gpt-oss:20b: fail (error) 2026-10-04 — The sentence claims the integrand is negative where it is negative, but for x<-2 the expression -2x-4 is actually positive. The sign analysis is incorrect, leading to an incorrect setup of the absolute value decomposition.
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.