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Area between curves

Problem 5.125 · medium

Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 6 x - 8 \) and \( \displaystyle y = - 3 x - 2 \).

\[ \left(6 - 3 x\right) \left(x - 1\right) = - 3 x^{2} + 9 x - 6 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = 1 and x = 2; between them y = -3*x**2 + 6*x - 8 is on top.
\[ \int\limits_{1}^{2} \left(- 3 x^{2} + 9 x - 6\right)\, dx = \frac{1}{2} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{1}{2} \)

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; solving -3x^2 + 6x - 8 = -3x - 2 yields x = 2 and x = 4/3, not 1 and 2. Consequently, the integrand and bounds are wrong.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-27 — The intersection points are incorrect; solving -3x^2 + 6x - 8 = -3x - 2 yields x = 2 and x = 4/3, not 1 and 2. Consequently, the integrand and bounds are wrong.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The intersection points are incorrect; solving -3x^2 + 6x - 8 = -3x - 2 yields x = 1 and x = 2, but substituting x=1 into the original equations gives y = -5 and y = -5, while x=2 gives y = -8 and y = -8. Wait, let's recheck. -3(1)^2 + 6(1) - 8 = -5. -3(1) - 2 = -5. Correct. -3(2)^2 + 6(2) - 8 = -12 + 12 - 8 = -8. -3(2) - 2 = -8. Correct. The intersection points are correct. However, the integrand in line 1 is -3x^2 + 9x - 6. Let's check top - bottom. Top is -3x^2 + 6x - 8. Bottom is -3x - 2. Top - Bottom = (-3x^2 + 6x - 8) - (-3x - 2) = -3x^2 + 9x - 6. This is correct. The integral of -3x^2 + 9x - 6 from 1 to 2 is [-x^3 + 4.5x^2 - 6x] from 1 to 2. At 2: -8 + 18 - 12 = -2. At 1: -1 + 4.5 - 6 = -2.5. Result: -2 - (-2.5) = 0.5. The calculation is correct. Why did I think it was wrong? Let me re-read the prompt. "Find the area...". The solution seems correct. Is there a subtle error? Line 1 says "-3*(x - 2)*(x - 1), -3*x**2 + 9*x - 6". This is just showing the factored form. Line 2 says "between them y = ... is on top". Let's check x=1.5. y1 = -3(2.25) + 9 - 8 = -6.75 + 1 = -5.75. y2 = -4.5 - 2 = -6.5. -5.75 > -6.5. So y1 is on top. Correct. The area is 1/2. The stated answer is 1/2. Everything seems correct. Wait, look at line 1 again. It is marked [equation, unchecked]. The prompt says "Each line... is either an EQUATION... or a SENTENCE". Line 1 is an equation line. Line 2 is a sentence. Line 3 is an equation line. The review is for the sentences. Line 2 is the only sentence. "They cross at x = 1 and x = 2; between them y = -3*x**2 + 6*x - 8 is on top." This sentence is factually correct. Is there any other sentence? No. So it should pass? Let me double check the intersection calculation again. -3x^2 + 6x - 8 = -3x - 2 => -3x^2 + 9x - 6 = 0 => x^2 - 3x + 2 = 0 => (x-1)(x-2)=0. Roots 1, 2. Correct. Top/Bottom check: Parabola opens downward. Line has slope -3. At x=1.5, parabola is higher. Correct. Integral value 1/2. Correct. Verdict: pass.
gpt-oss:20b: pass 2026-09-27

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 2026-09-27 with SymPy 1.14.0.