Integrals of absolute values
Problem 4.284 · easy
Evaluate \( \displaystyle \int_{-3}^{2} \left| - x^{2} - x \right| dx \).
- \[ - x^{2} - x = - x \left(x + 1\right) \]The integrand is zero at x = -1, x = 0.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).
- \[ \int\limits_{-1}^{0} \left(- x^{2} - x\right)\, dx + \int\limits_{0}^{2} \left(x^{2} + x\right)\, dx + \int\limits_{-3}^{-1} \left(x^{2} + x\right)\, dx = \frac{19}{2} \]Integrate each piece and add.✓ Proved
Answer \( \frac{19}{2} \)
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, determines the sign of the integrand in each interval, and splits the integral accordingly. The final calculation is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the roots, determines the sign of the integrand in each interval, and splits the integral accordingly. The final calculation is correct.gpt-oss:20b: inconclusive 2026-10-04 — reviewer returned a non-objectqwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer returned a non-objectgpt-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.