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Area between curves

Problem 5.120 · medium

Find the area of the region bounded by \( \displaystyle y = - x^{2} + 7 x - 9 \) and \( \displaystyle y = x - 1 \).

\[ \left(4 - x\right) \left(x - 2\right) = - x^{2} + 6 x - 8 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = 2 and x = 4; between them y = -x**2 + 7*x - 9 is on top.
Reviewed
\[ \int\limits_{2}^{4} \left(- x^{2} + 6 x - 8\right)\, dx = \frac{4}{3} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{4}{3} \)

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

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-27 — 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 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The setup in line 1 calculates the difference as -x^2 + 6x - 8, but the actual difference (top - bottom) is (-x^2 + 7x - 9) - (x - 1) = -x^2 + 6x - 8. Wait, let me re-calculate. (-x^2 + 7x - 9) - (x - 1) = -x^2 + 6x - 8. The intersection points are found by -x^2 + 6x - 8 = 0 => x^2 - 6x + 8 = 0 => (x-2)(x-4)=0. So x=2, 4. The integral of -x^2 + 6x - 8 from 2 to 4 is indeed 4/3. The solution is actually correct. Why did I think it was wrong? Let me double check the problem statement. y = -x^2 + 7x - 9 and y = x - 1. Difference: -x^2 + 7x - 9 - x + 1 = -x^2 + 6x - 8. Roots: 2, 4. Integral: [-x^3/3 + 3x^2 - 8x] from 2 to 4. At 4: -64/3 + 48 - 32 = -64/3 + 16 = -64/3 + 48/3 = -16/3. At 2: -8/3 + 12 - 16 = -8/3 - 4 = -8/3 - 12/3 = -20/3. Result: -16/3 - (-20/3) = 4/3. The solution is correct. The sentence in line 2 says 'between them y = -x**2 + 7*x - 9 is on top'. Let's check x=3. Top: -9 + 21 - 9 = 3. Bottom: 3 - 1 = 2. 3 > 2. Correct. The solution is sound.
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.