Integrals of absolute values
Problem 4.586 · easy
Evaluate \( \displaystyle \int_{1}^{8} \left| x^{2} - 9 x + 18 \right| dx \).
- \[ x^{2} - 9 x + 18 = \left(x - 6\right) \left(x - 3\right) \]The integrand is zero at x = 3, x = 6.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).Reviewed
- \[ \int\limits_{3}^{6} \left(- x^{2} + 9 x - 18\right)\, dx + \int\limits_{1}^{3} \left(x^{2} - 9 x + 18\right)\, dx + \int\limits_{6}^{8} \left(x^{2} - 9 x + 18\right)\, dx = \frac{131}{6} \]Integrate each piece and add.✓ Proved
Answer \( \frac{131}{6} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the roots of the quadratic, determines the sign of the integrand in each sub-interval, and splits the integral accordingly. The final calculation is correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the roots of the quadratic, determines the sign of the integrand in each sub-interval, and splits the integral accordingly. The final calculation is correct.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — 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 result is accurate.
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.