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Area between curves

Problem 5.69 · medium Video

Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 15 x - 21 \) and \( \displaystyle y = x + 3 \).

\[ \left(8 - 2 x\right) \left(x - 3\right) = - 2 x^{2} + 14 x - 24 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = 3 and x = 4; between them y = -2*x**2 + 15*x - 21 is on top.
\[ \int\limits_{3}^{4} \left(- 2 x^{2} + 14 x - 24\right)\, dx = \frac{1}{3} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{1}{3} \)

The curves y = -2x² + 15x - 21 and y = x + 3 between x = 3 and x = 4. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 1/3.

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) is -2x^2 + 14x - 24, but the problem states the parabola is -2x^2 + 15x - 21. The co

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The setup in line 1 is incorrect: the difference (top - bottom) is -2x^2 + 14x - 24, but the problem states the parabola is -2x^2 + 15x - 21. The co
gpt-oss:20b: pass 2026-09-28
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: it factors the difference as -2(x-4)(x-3), which expands to -2x^2 + 14x - 24. However, the actual difference (top
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The setup in line 1 is incorrect: the difference (top - bottom) is -2x^2 + 14x - 24, but the problem states the parabola is y = -2x^2 + 15x - 21. The linear term in the difference should be 14x only if the parabola was -2x^2 + 14x - 24, which it is not. The intersection points and the resulting integral are based on the wrong function.
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 2026-09-28 with SymPy 1.14.0.