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Area between curves

Problem 5.97 · medium Video

Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 18 x - 21 \) and \( \displaystyle y = 3 x - 3 \).

\[ \left(9 - 3 x\right) \left(x - 2\right) = - 3 x^{2} + 15 x - 18 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = 2 and x = 3; between them y = -3*x**2 + 18*x - 21 is on top.
\[ \int\limits_{2}^{3} \left(- 3 x^{2} + 15 x - 18\right)\, dx = \frac{1}{2} \]
Area = ∫ (top − bottom) dx.✓ Proved

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

The curves y = -3x² + 18x - 21 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 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. How we verify

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: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The setup in line 1 is incorrect: the difference (top - bottom) simplifies to -3x^2 + 15x - 18, but the problem states the parabola is y = -3x^2 + 18

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The setup in line 1 is incorrect: the difference (top - bottom) simplifies to -3x^2 + 15x - 18, but the problem states the parabola is y = -3x^2 + 18
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The intersection points are incorrect; solving -3x^2 + 18x - 21 = 3x - 3 yields x = 1 and x = 2, not 2 and 3. Consequently, the integrand and bounds are wrong.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The integrand in step 1 is incorrect; the difference (top - bottom) should be -3x^2 + 15x - 18, but the problem states the parabola is y = -3x^2 + 18
gpt-oss:20b: pass 2026-09-26

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.