Integrals of absolute values
Problem 4.887 · 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: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly states that the integrand is zero at x = -2, implying this is the only zero, but fails to explicitly split the integral at this point in the text before jumping to the calculation. More critically, the sentence 'Between those zeros' is nonsensical as there is only one zero for a linear function, and the logic for splitting the absolute value is poorly articulated, potentially confusing the student about why the split occurs at -2.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution incorrectly states that the integrand is zero at x = -2, implying this is the only zero, but fails to explicitly split the integral at this point in the text before jumping to the calculation. More critically, the sentence 'Between those zeros' is nonsensical as there is only one zero for a linear function, and the logic for splitting the absolute value is poorly articulated, potentially confusing the student about why the split occurs at -2.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the split point at x = -2, applies the definition of absolute value to split the integral into two parts with correct signs, and computes the correct final value.
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 2026-10-11 with SymPy 1.14.0.