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Area between curves

Problem 5.119 · medium

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

\[ \left(6 - 3 x\right) \left(x + 1\right) = - 3 x^{2} + 3 x + 6 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -1 and x = 2; between them y = -3*x**2 + 4*x + 4 is on top.
\[ \int\limits_{-1}^{2} \left(- 3 x^{2} + 3 x + 6\right)\, dx = \frac{27}{2} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{27}{2} \)

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: fail (error) — The intersection points are incorrect; solving -3x^2 + 4x + 4 = x - 2 yields x = 2 and x = 4/3, not x = -1. Consequently, the integration bounds and the integrand used in step 3 are wrong.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-27 — The intersection points are incorrect; solving -3x^2 + 4x + 4 = x - 2 yields x = 2 and x = 4/3, not x = -1. Consequently, the integration bounds and the integrand used in step 3 are wrong.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the intersection points and the relative position of the curves. The integrand represents the correct difference (top minus bottom), and the integration yields the correct area.
gpt-oss:20b: pass 2026-09-27

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 2026-09-27 with SymPy 1.14.0.