Area between curves
Problem 5.145 · medium
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} - x + 15 \) and \( \displaystyle y = 3 - 3 x \).
- \[ \left(6 - 2 x\right) \left(x + 2\right) = - 2 x^{2} + 2 x + 12 \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = -2 and x = 3; between them y = -2*x**2 - x + 15 is on top.
- \[ \int\limits_{-2}^{3} \left(- 2 x^{2} + 2 x + 12\right)\, dx = \frac{125}{3} \]Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{125}{3} \)
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: fail (error) — The setup in line 1 is incorrect: the difference between the curves is -2x^2 + 2x + 12, but the problem states the quadratic is -2x^2 - x + 15. The roots of the actual difference (-2x^2 + 2x + 12) are indeed -2 and 3, but the integrand used does not match the functions given in the problem statement.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The setup in line 1 is incorrect: the difference between the curves is -2x^2 + 2x + 12, but the problem states the quadratic is -2x^2 - x + 15. The roots of the actual difference (-2x^2 + 2x + 12) are indeed -2 and 3, but the integrand used does not match the functions given in the problem statement.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The setup in line 1 calculates the difference as -2x^2 + 2x + 12, but the actual difference (top - bottom) is (-2x^2 - x + 15) - (3 - 3x) = -2x^2 +gpt-oss:20b: pass 2026-09-28
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 2026-09-28 with SymPy 1.14.0.