Area between curves
Problem 5.196 · medium
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - 3 x + 9 \) and \( \displaystyle y = 3 x \).
- \[ \left(3 - 3 x\right) \left(x + 3\right) = - 3 x^{2} - 6 x + 9 \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = -3 and x = 1; between them y = -3*x**2 - 3*x + 9 is on top.
- \[ \int\limits_{-3}^{1} \left(- 3 x^{2} - 6 x + 9\right)\, dx = 32 \]Area = ∫ (top − bottom) dx.✓ Proved
Answer \( 32 \)
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 |top − bottom| gives the same area |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The setup incorrectly defines the integrand as -3x^2 - 6x + 9, which corresponds to y = -3x^2 - 6x + 9, not the given curve y = -3x^2 - 3x + 9. The c
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-03 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The setup incorrectly defines the integrand as -3x^2 - 6x + 9, which corresponds to y = -3x^2 - 6x + 9, not the given curve y = -3x^2 - 3x + 9. The cgpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: inconclusive 2026-10-03 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The setup in line 1 incorrectly defines the integrand as -3x^2 - 6x + 9, which corresponds to subtracting 3x from -3x^2 - 6x + 9, not the given curvegpt-oss:20b: pass 2026-10-03
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/area_between_curves, checked 2026-10-03 with SymPy 1.14.0.