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Area between curves

Problem 5.173 · medium Video

Find the area of the region bounded by \( \displaystyle y = 15 - 3 x^{2} \) and \( \displaystyle y = - 3 x - 3 \).

\[ \left(9 - 3 x\right) \left(x + 2\right) = - 3 x^{2} + 3 x + 18 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -2 and x = 3; between them y = 15 - 3*x**2 is on top.
Reviewed
\[ \int\limits_{-2}^{3} \left(- 3 x^{2} + 3 x + 18\right)\, dx = \frac{125}{2} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{125}{2} \)

The curves y = 15 - 3x² and y = -3x - 3 between x = -2 and x = 3. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 125/2.

✓ 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 upper and lower functions, 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 upper and lower functions, 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, verifies the relative position of the curves, and computes the definite integral accurately.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The intersection points are correctly identified, the relative position of the curves is correct on the interval, and the integration yields the correct area.
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.