Integrals of absolute values
Problem 4.894 · easy
Evaluate \( \displaystyle \int_{-1}^{6} \left| - x^{2} + 4 x - 3 \right| dx \).
- \[ - x^{2} + 4 x - 3 = \left(3 - x\right) \left(x - 1\right) \]The integrand is zero at x = 1, x = 3.✓ Proved
- Between those zeros the sign does not change; where it is negative, |p(x)| = −p(x).Reviewed
- \[ \int\limits_{1}^{3} \left(- x^{2} + 4 x - 3\right)\, dx + \int\limits_{-1}^{1} \left(x^{2} - 4 x + 3\right)\, dx + \int\limits_{3}^{6} \left(x^{2} - 4 x + 3\right)\, dx = 26 \]Integrate each piece and add.✓ Proved
Answer \( 26 \)
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 quadratic, determines the sign of the polynomial on the sub-intervals, and sets up the piecewise integration with the correct signs for the absolute value. The final calculation is correct.
Every verdict on record (4)
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 on the sub-intervals, and sets up the piecewise integration with the correct signs for the absolute value. The final calculation is correct.gpt-oss:20b: fail (error) 2026-10-11 — The computed integral equals 23, not 26. The piecewise integration is correct in principle, but the arithmetic or bounds used in the final step are incorrect.qwen3.6:27b-mlx: pass 2026-10-11 — 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 2026-10-11 with SymPy 1.14.0.