Integrals of absolute values
Problem 4.584 · easy
Evaluate \( \displaystyle \int_{-6}^{3} \left| - x^{2} - 3 x \right| dx \).
- \[ - x^{2} - 3 x = - x \left(x + 3\right) \]The integrand is zero at x = -3, x = 0.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).Reviewed
- \[ \int\limits_{-3}^{0} \left(- x^{2} - 3 x\right)\, dx + \int\limits_{0}^{3} \left(x^{2} + 3 x\right)\, dx + \int\limits_{-6}^{-3} \left(x^{2} + 3 x\right)\, dx = \frac{99}{2} \]Integrate each piece and add.✓ Proved
Answer \( \frac{99}{2} \)
Lines: 2 proved, 1 reviewed. 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 | 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 polynomial, determines the sign of the integrand in each sub-interval, and sets up the piecewise integration correctly. The final arithmetic 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 polynomial, determines the sign of the integrand in each sub-interval, and sets up the piecewise integration correctly. The final arithmetic is correct.gpt-oss:20b: fail (misleading) 2026-10-07 — The sentence about the sign of the integrand is vague and does not specify the intervals where the function is positive or negative. It could mislead a student into thinking the sign does not change between the zeros, whereas it actually does change on each subinterval. The setup and algebraic steps are correct, but the explanation is incomplete and potentially confusing.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the roots of the polynomial, determines the sign of the integrand on each sub-interval, and sets up the piecewise integration correctly. The final arithmetic result is correct.
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.