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Area between curves

Problem 5.191 · medium

Find the area of the region bounded by \( \displaystyle y = - x^{2} - 4 x + 3 \) and \( \displaystyle y = - 3 x - 3 \).

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

Answer \( \frac{125}{6} \)

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

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-03 — 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 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The setup incorrectly identifies the intersection points as x = -3 and x = 2. Solving -x^2 - 4x + 3 = -3x - 3 yields x^2 + x - 6 = 0, which factors to (x+3)(x-2)=0, so the roots are indeed -3 and 2. Wait, let me re-check the subtraction. Top - Bottom = (-x^2 - 4x + 3) - (-3x - 3) = -x^2 - x + 6. The roots of -x^2 - x + 6 = 0 are x = 2 and x = -3. So the intersection points are correct. The integral calculation is also correct. Why did I think it was wrong? Let me re-read carefully. Ah, looking at line 1: `-(x - 2)*(x + 3) = -x**2 - x + 6`. This is algebraically correct. Line 2: "They cross at x = -3 and x = 2". Correct. "between them y = -x**2 - 4*x + 3 is on top". Let's check a point, e.g., x=0. y1 = 3, y2 = -3. 3 > -3, so the parabola is on top. Correct. Line 3: Integral of (-x^2 - x + 6) from -3 to 2. Antiderivative: -x^3/3 - x^2/2 + 6x. At 2: -8/3 - 2 + 12 = -8/3 + 10 = 22/3. At -3: -(-27)/3 - 9/2 - 18 = 9 - 4.5 - 18 = -13.5 = -27/2. Difference: 22/3 - (-27/2) = 44/6 + 81/6 = 125/6. The solution is actually correct. I will mark it as pass.
gpt-oss:20b: pass 2026-10-03

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 2026-10-03 with SymPy 1.14.0.