Area between curves
Problem 5.194 · medium
Find the area of the region bounded by \( \displaystyle y = - x^{2} + 3 x + 4 \) and \( \displaystyle y = x + 1 \).
- \[ \left(3 - x\right) \left(x + 1\right) = - x^{2} + 2 x + 3 \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = -1 and x = 3; between them y = -x**2 + 3*x + 4 is on top.
- \[ \int\limits_{-1}^{3} \left(- x^{2} + 2 x + 3\right)\, dx = \frac{32}{3} \]Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{32}{3} \)
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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The setup in step 1 is incorrect: the difference between the curves is (-x^2 + 3x + 4) - (x + 1) = -x^2 + 2x + 3, but the equation shown factors -x^
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 in step 1 is incorrect: the difference between the curves is (-x^2 + 3x + 4) - (x + 1) = -x^2 + 2x + 3, but the equation shown factors -x^gpt-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 is incorrect: it computes the difference as (-x^2 + 2x + 3) instead of the correct (top - bottom) = (-x^2 + 3x + 4) - (x + 1) =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.