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Area between curves

Problem 5.96 · medium

Find the area of the region bounded by \( \displaystyle y = - x^{2} - 6 x - 9 \) and \( \displaystyle y = - x - 3 \).

\[ \left(- x - 2\right) \left(x + 3\right) = - x^{2} - 5 x - 6 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -3 and x = -2; between them y = -x**2 - 6*x - 9 is on top.
\[ \int\limits_{-3}^{-2} \left(- x^{2} - 5 x - 6\right)\, dx = \frac{1}{6} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{1}{6} \)

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; the curves intersect at x = -3 and x = -1, not x = -2. Consequently, the integration bounds and the resulting area are wrong.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-26 — The intersection points are incorrect; the curves intersect at x = -3 and x = -1, not x = -2. Consequently, the integration bounds and the resulting area are wrong.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The intersection points are incorrect; solving -x^2 - 6x - 9 = -x - 3 yields x = -2 and x = -1, not -3 and -2. Consequently, the integration bounds and the integrand (which should be -x^2 - 5x - 6) are based on wrong roots, although the integrand expression itself matches the difference of the functions provided in line 1, the bounds are wrong.
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 2026-09-26 with SymPy 1.14.0.