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Area between curves

Problem 5.143 · medium

Find the area of the region bounded by \( \displaystyle y = - x^{2} + 4 x \) and \( \displaystyle y = 2 x - 3 \).

\[ \left(3 - x\right) \left(x + 1\right) = - x^{2} + 2 x + 3 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -1 and x = 3; between them y = -x**2 + 4*x 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: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The intersection points are calculated incorrectly. Solving -x^2 + 4x = 2x - 3 yields x^2 - 2x - 3 = 0, so the roots are x = 3 and x = -1. Wait, the

Every verdict on record (4)

qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The intersection points are calculated incorrectly. Solving -x^2 + 4x = 2x - 3 yields x^2 - 2x - 3 = 0, so the roots are x = 3 and x = -1. Wait, the
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the intersection points, determines the correct upper and lower functions on the interval, and computes the definite integral accurately.
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.