Area between curves
Problem 5.169 · medium
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 12 x - 27 \) and \( \displaystyle y = - 2 x - 3 \).
- \[ \left(8 - 2 x\right) \left(x - 3\right) = - 2 x^{2} + 14 x - 24 \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = 3 and x = 4; between them y = -2*x**2 + 12*x - 27 is on top.
- \[ \int\limits_{3}^{4} \left(- 2 x^{2} + 14 x - 24\right)\, dx = \frac{1}{3} \]Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{1}{3} \)
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 calculates the difference for y = -2x^2 + 14x - 24, but the problem specifies y = -2x^2 + 12x - 27. Consequently, the intersection points (3 and 4) and the resulting area (1/3) are incorrect for the given functions.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The setup in line 1 calculates the difference for y = -2x^2 + 14x - 24, but the problem specifies y = -2x^2 + 12x - 27. Consequently, the intersection points (3 and 4) and the resulting area (1/3) are incorrect for the given functions.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (error) 2026-09-29 — The setup in line 1 computes the difference between y = -2x^2 + 14x - 24 and y = -2x - 3, but the problem specifies y = -2x^2 + 12x - 27. The coefficients of the quadratic are incorrect, leading to wrong intersection points and area.gpt-oss:20b: pass 2026-09-29
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 2026-09-29 with SymPy 1.14.0.