Area between curves
Problem 5.195 · medium
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 11 x + 3 \) and \( \displaystyle y = 3 x + 3 \).
- \[ - 2 x \left(x - 4\right) = - 2 x^{2} + 8 x \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = 0 and x = 4; between them y = -2*x**2 + 11*x + 3 is on top.Reviewed
- \[ \int\limits_{0}^{4} \left(- 2 x^{2} + 8 x\right)\, dx = \frac{64}{3} \]Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{64}{3} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 |top − bottom| gives the same area |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the intersection points, determines the correct upper and lower functions, and computes the definite integral accurately.gpt-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.