Integrals of absolute values
Problem 4.591 · easy
Evaluate \( \displaystyle \int_{-3}^{-1} \left| - x - 2 \right| dx \).
- \[ - x - 2 \]The integrand is zero at x = -2.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).
- \[ \int\limits_{-3}^{-2} \left(- x - 2\right)\, dx + \int\limits_{-2}^{-1} \left(x + 2\right)\, dx = 1 \]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: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution claims the integrand is zero at x = -2, but -(-2) - 2 = 0 is correct, however, the sign analysis in step 2 is flawed. For x in [-3, -2), -x-2 is positive (e.g., at x=-3, -(-3)-2=1), so | -x-2 | = -x-2. For x in (-2, -1], -x-2 is negative (e.g., at x=-1, -(-1)-2=-1), so | -x-2 | = -(-x-2) = x+2. The setup in step 3 correctly uses -x-2 for [-3, -2] and x+2 for [-2, -1], but the sentence in step 2 says 'where it is negative, |p(x)| = -p(x)' which is true, but implies the first interval is where it is negative? No, it says 'Between those zeros' (singular zero) the sign does not change. This is confusing phrasing for a single root. More importantly, the sentence 'Between those zeros' implies multiple zeros, but there is only one. The logic is garbled. Let's look closer. Step 1 says 'The integrand is zero at x = -2'. Step 2 says 'Between those zeros...'. There is only one zero. This is a factual error in the text. Also, for x < -2, -x-2 > 0, so | -x-2 | = -x-2. The solution integrates -x-2 from -3 to -2. This is correct. For x > -2, -x-2 < 0, so | -x-2 | = x+2. The solution integrates x+2 from -2 to -1. This is correct. The calculation is correct. The error is in the sentence 'Between those zeros'. There is only one zero. This is misleading/confusing but is it an 'error' in the math? It's a linguistic error. However, usually 'error' covers logical flaws. Let's check the sign again. x=-3: |-3-2|? No, |-(-3)-2| = |3-2| = 1. Formula -x-2 gives 1. Correct. x=-1: |-(-1)-2| = |1-2| = |-1| = 1. Formula x+2 gives 1. Correct. The math is right. The sentence is wrong because it refers to 'those zeros' (plural) when there is only one. This is a 'misleading' or 'style' issue? Or 'error'? If a student reads 'between those zeros', they might look for another zero. It's a factual statement error. I will mark it as error because the premise of the sentence is false (there are not multiple zeros).
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 2026-10-07 with SymPy 1.14.0.