Integrals of absolute values
Problem 4.889 · easy
Evaluate \( \displaystyle \int_{0}^{7} \left| - x^{2} + 6 x - 8 \right| dx \).
- \[ - x^{2} + 6 x - 8 = \left(4 - x\right) \left(x - 2\right) \]The integrand is zero at x = 2, x = 4.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).
- \[ \int\limits_{2}^{4} \left(- x^{2} + 6 x - 8\right)\, dx + \int\limits_{0}^{2} \left(x^{2} - 6 x + 8\right)\, dx + \int\limits_{4}^{7} \left(x^{2} - 6 x + 8\right)\, dx = 26 \]Integrate each piece and add.✓ Proved
Answer \( 26 \)
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: fail (misleading) — The sentence claims the sign does not change between the zeros, which is false; the polynomial is positive on (2,4) and negative elsewhere. The subsequent integrals are correct, but the statement misleads a student about sign behavior.qwen3.6:27b-mlx: pass — The solution correctly identifies the roots of the quadratic, determines the sign of the polynomial in each interval, and sets up the piecewise integration of the absolute value function accurately. The final calculation is correct.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-11 — The sentence claims the sign does not change between the zeros, which is false; the polynomial is positive on (2,4) and negative elsewhere. The subsequent integrals are correct, but the statement misleads a student about sign behavior.qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the roots of the quadratic, determines the sign of the polynomial in each interval, and sets up the piecewise integration of the absolute value function accurately. The final calculation is correct.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the roots of the quadratic, determines the sign of the polynomial in each interval, and sets up the piecewise integration with the correct signs for the absolute value. The final calculation 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 2026-10-11 with SymPy 1.14.0.