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Area between curves

Problem 5.149 · medium

Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 5 x + 10 \) and \( \displaystyle y = 2 - x \).

\[ \left(8 - 2 x\right) \left(x + 1\right) = - 2 x^{2} + 6 x + 8 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -1 and x = 4; between them y = -2*x**2 + 5*x + 10 is on top.
Reviewed
\[ \int\limits_{-1}^{4} \left(- 2 x^{2} + 6 x + 8\right)\, dx = \frac{125}{3} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{125}{3} \)

Lines: 2 proved, 1 reviewed. 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	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 on the interval, and computes the definite integral accurately.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the intersection points, determines the correct upper and lower functions on the interval, and computes the definite integral accurately.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The setup in step 1 incorrectly calculates the difference between the curves. The actual difference is (-2x^2 + 5x + 10) - (2 - x) = -2x^2 + 6x + 8, but the factorization provided (-2*(x - 4)*(x + 1)) expands to -2x^2 + 6x + 8, which matches the difference, not the original top curve. However, the roots of -2x^2 + 6x + 8 are indeed -1 and 4. Wait, let's re-read carefully. Step 1 says 'Set the curves equal: top - bottom, factored'. The expression -2*x**2 + 6*x + 8 is indeed the integrand. The factorization -2*(x-4)*(x+1) is correct for that integrand. The roots are -1 and 4. Step 2 claims the curves cross at -1 and 4. This is correct. Step 2 claims the quadratic is on top. This is correct for a downward parabola between roots. Step 3 integrates the correct difference. The solution is actually correct. Why did I think it was wrong? Let me re-verify the factorization. -2(x-4)(x+1) = -2(x^2 - 3x - 4) = -2x^2 + 6x + 8. The difference is (-2x^2 + 5x + 10) - (2 - x) = -2x^2 + 6x + 8. Everything is correct. The verdict should be pass.
gpt-oss:20b: pass 2026-09-28

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.