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Integrals of absolute values

Problem 4.291 · easy

Evaluate \( \displaystyle \int_{-5}^{-1} \left| - x - 2 \right| dx \).

\[ - x - 2 \]
The integrand is zero at x = -2.✓ Proved
Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).
\[ \int\limits_{-2}^{-1} \left(x + 2\right)\, dx + \int\limits_{-5}^{-2} \left(- x - 2\right)\, dx = 5 \]
Integrate each piece and add.✓ Proved

Answer \( 5 \)

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: inconclusive — reviewer returned a non-object
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (4)

qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer returned a non-object
gpt-oss:20b: inconclusive 2026-10-04 — reviewer returned a non-object
qwen3.6:27b-mlx: fail (style) 2026-10-04 — [domain objection, downgraded to style] The solution incorrectly identifies the zero of the integrand as x = -2, whereas -x - 2 = 0 implies x = -2 is correct, but the sign analysis is flawed. For x in [-5, -2), -x - 2 is positive (e.g., at x=-5, -(-5)-2=3), 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 solution swaps the expressions for the two intervals: it integrates x+2 on [-2, -1] (correct) but integrates -x-2 on [-5, -2] (correct expression, but the text says 'where it is negative, |p(x)| = -p(x)' implying the first interval was negative, which is false; actually the first interval is positive). Wait, let's re-read carefully. Integrand: |-x-2|. Zero at x=-2. Interval 1: [-5, -2]. Test x=-3: |-(-3)-2| = |3-2| = 1. Expression -x-2 at x=-3 is 1. So |-x-2| = -x-2. This is positive. Interval 2: [-2, -1]. Test x=-1.5: |-(-1.5)-2| = |1.5-2| = |-0.5| = 0.5. Expression -x-2 at x=-1.5 is -0.5. So |-x-2| = -(-x-2) = x+2. This is negative inside the absolute value. The solution says: 'Between those zeros... where it is negative, |p(x)| = -p(x)'. This is a general statement. Then it computes: Integral(x + 2, (x, -2, -1)) + Integral(-x - 2, (x, -5, -2)). Let's check the pieces. Piece 1: x from -2 to -1. Integrand x+2. This corresponds to the region where -x-2 is negative. Correct. Piece 2: x from -5 to -2. Integrand -x-2. This corresponds to the region where -x-2 is positive. Correct. So the calculation is actually correct. Why did I think it was wrong? Let's re-read the sentence: 'Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).' This sentence is slightly ambiguous. It implies there is only one 'zero' mentioned in line 1? Line 1 says 'The integrand is zero at x = -2'. It does not mention another zero. The absolute value function |-x-2| has only one zero at x=-2. The phrase 'Between those zeros' is confusing because there is only one zero. Usually, this phrasing is used when splitting an interval based on roots of the function inside the absolute value. Here, the split is at the single root x=-2. The phrase 'Between those zeros' suggests multiple zeros, which is misleading or incorrect contextually if it implies the interval is bounded by zeros. However, the integration limits are -5 and -1, which are not zeros. Actually, the bigger issue is the phrasing 'Between those zeros'. There is only one zero, x=-2. The interval of integration is [-5, -1]. The split point is x=-2. The sentence 'Between those zeros' is nonsensical if it refers to the bounds of the integral or if it implies there are two zeros. If it refers to the pieces of the domain split by the zero, it's poorly phrased. However, looking closer at the provided solution text: 1. '-x - 2 = -x - 2 -- The integrand is zero at x = -2.' 2. 'Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).' The phrase 'Between those zeros' is the error. There is only one zero. You cannot be 'between' a single zero. The student likely confused this with a problem involving a quadratic or multiple roots, or simply used a template incorrectly. This is a 'misleading' or 'error' in the explanation. Let's check the calculation again just to be sure. Integral from -5 to -2 of (-x-2) dx. Antiderivative -x^2/2 - 2x. At -2: -2 - (-4) = 2. At -5: -12.5 - (-10) = -2.5. Result: 2 - (-2.5) = 4.5. Integral from -2 to -1 of (x+2) dx. Antiderivative x^2/2 + 2x. At -1: 0.5 - 2 = -1.5. At -2: 2 - 4 = -2. Result: -1.5 - (-2) = 0.5. Total: 4.5 + 0.5 = 5. The calculation is correct. The error is purely in the sentence 'Between those zeros'. This is a significant logical flaw in the explanation. It teaches the student that absolute value splits happen 'between zeros' (plural) or that the current problem has multiple zeros. Verdict: Fail. Severity: Misleading (or Error in reasoning description). I will call it 'error' because the premise 'those zeros' is factually false for this function.
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.