Integrals of absolute values
Problem 4.1024 · easy
Evaluate \( \displaystyle \int_{-6}^{0} \left| - x^{2} - 5 x - 6 \right| dx \).
- \[ - x^{2} - 5 x - 6 = \left(- x - 2\right) \left(x + 3\right) \]The integrand is zero at x = -3, x = -2.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).
- \[ \int\limits_{-3}^{-2} \left(- x^{2} - 5 x - 6\right)\, dx + \int\limits_{-2}^{0} \left(x^{2} + 5 x + 6\right)\, dx + \int\limits_{-6}^{-3} \left(x^{2} + 5 x + 6\right)\, dx = \frac{55}{3} \]Integrate each piece and add.✓ Proved
Answer \( \frac{55}{3} \)
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-objectqwen3.6:27b-mlx: pass — The solution correctly identifies the roots of the quadratic, determines the sign of the integrand in each sub-interval, and sets up the piecewise integration correctly. The final calculation is accurate.
Every verdict on record (4)
gpt-oss:20b: inconclusive 2026-10-13 — reviewer returned a non-objectqwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the roots of the quadratic, determines the sign of the integrand in each sub-interval, and sets up the piecewise integration correctly. The final calculation is accurate.gpt-oss:20b: fail (error) 2026-10-13 — The integrand is positive on (-3,-2), so |p(x)| should be p(x) there, not -p(x). The piecewise integration in step 3 uses the wrong sign for that interval, leading to an incorrect result.qwen3.6:27b-mlx: inconclusive 2026-10-13 — reviewer returned a non-object
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 2026-10-13 with SymPy 1.14.0.