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Area between curves

Problem 5.10 · medium Video

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

\[ \left(2 - x\right) \left(x + 2\right) = 4 - x^{2} \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -2 and x = 2; between them y = -x**2 + x + 2 is on top.
\[ \int\limits_{-2}^{2} \left(4 - x^{2}\right)\, dx = \frac{32}{3} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{32}{3} \)

The curves y = -x² + x + 2 and y = x - 2 between x = -2 and x = 2. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 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. How we verify

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 calculated incorrectly; the curves intersect at x = -1 and x = 3, not -2 and 2. Additionally, the integrand in line 1 omits the linear term from the subtraction, resulting in an incorrect area calculation.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The intersection points are calculated incorrectly; the curves intersect at x = -1 and x = 3, not -2 and 2. Additionally, the integrand in line 1 omits the linear term from the subtraction, resulting in an incorrect area calculation.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The intersection points are incorrect; the curves intersect at x = -1 and x = 3, not -2 and 2. Consequently, the integrand is also wrong (it should be -x^2 + 2x + 4, not 4 - x^2).
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The intersection points are incorrect; solving -x^2 + x + 2 = x - 2 yields x = +/- 2*sqrt(2), not +/- 2. Consequently, the integrand is also wrong (it should be -x^2 + 4, but the bounds are the primary error leading to the wrong area).
gpt-oss:20b: pass 2026-09-26

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.