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Area between curves

Problem 5.3 · medium Video

Find the area of the region bounded by \( \displaystyle y = - x^{2} - x + 10 \) and \( \displaystyle y = 2 - 3 x \).

\[ \left(4 - x\right) \left(x + 2\right) = - x^{2} + 2 x + 8 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -2 and x = 4; between them y = -x**2 - x + 10 is on top.
Reviewed
\[ \int\limits_{-2}^{4} \left(- x^{2} + 2 x + 8\right)\, dx = 36 \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( 36 \)

The curves y = -x² - x + 10 and y = 2 - 3x between x = -2 and x = 4. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 36.

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the intersection points and the relative position of the curves. The integrand and bounds are correct, leading to the right answer.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the intersection points and the relative position of the curves. The integrand and bounds are correct, leading to the right answer.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the intersection points, determines the correct integrand (top curve minus bottom curve), and computes the definite integral accurately.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the intersection points, determines the correct integrand (top curve minus bottom curve), and computes the definite integral accurately.
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.