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Area between curves

Problem 5.167 · medium Video

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

\[ \left(4 - x\right) \left(x + 3\right) = - x^{2} + x + 12 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -3 and x = 4; between them y = -x**2 + 2*x + 10 is on top.
Reviewed
\[ \int\limits_{-3}^{4} \left(- x^{2} + x + 12\right)\, dx = \frac{343}{6} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{343}{6} \)

The curves y = -x² + 2x + 10 and y = x - 2 between x = -3 and x = 4. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 343/6.

✓ 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, determines the correct integrand (top curve minus bottom curve), and computes the definite integral accurately.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-04 — 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 2026-10-04
qwen3.6:27b-mlx: pass 2026-09-29 — 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 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — 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 2026-09-29

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 2026-10-04 with SymPy 1.14.0.