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Area between curves

Problem 5.121 · medium

Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 2 x + 1 \) and \( \displaystyle y = 2 x - 1 \).

\[ \left(2 - 2 x\right) \left(x + 1\right) = 2 - 2 x^{2} \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -1 and x = 1; between them y = -2*x**2 + 2*x + 1 is on top.
\[ \int\limits_{-1}^{1} \left(2 - 2 x^{2}\right)\, dx = \frac{8}{3} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{8}{3} \)

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: fail (error) — The intersection points are incorrect; solving -2x^2 + 2x + 1 = 2x - 1 yields x = ±sqrt(2), not ±1. Consequently, the integrand is also wrong (it should be -2x^2 + 2, not 2 - 2x^2).

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-27 — The intersection points are incorrect; solving -2x^2 + 2x + 1 = 2x - 1 yields x = ±sqrt(2), not ±1. Consequently, the integrand is also wrong (it should be -2x^2 + 2, not 2 - 2x^2).
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The intersection points are incorrect; solving -2x^2 + 2x + 1 = 2x - 1 yields x = ±1/sqrt(2), not ±1. Consequently, the integrand and bounds are wrong.
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.