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Area between curves

Problem 5.15 · medium Video

Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 7 x - 8 \) and \( \displaystyle y = - 3 x \).

\[ \left(8 - 2 x\right) \left(x - 1\right) = - 2 x^{2} + 10 x - 8 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = 1 and x = 4; between them y = -2*x**2 + 7*x - 8 is on top.
\[ \int\limits_{1}^{4} \left(- 2 x^{2} + 10 x - 8\right)\, dx = 9 \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( 9 \)

The curves y = -2x² + 7x - 8 and y = -3x between x = 1 and x = 4. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 9.

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: fail (error) — The intersection points are incorrect. Solving -2x^2 + 7x - 8 = -3x yields -2x^2 + 10x - 8 = 0, which factors to -2(x-1)(x-4)=0, so the roots are indeed 1 and 4. Wait, let me re-check the problem statement vs the solution. The problem is y = -2x^2 + 7x - 8 and y = -3x. The difference is (-2x^2 + 7x - 8) - (-3x) = -2x^2 + 10x - 8. The roots of -2x^2 + 10x - 8 = 0 are x = 1 and x = 4. The integral of -2x^2 + 10x - 8 from 1 to 4 is indeed 9. The solution seems correct. Why did I think it was wrong? Let me re-read carefully. Ah, the problem says y = -2x^2 + 7x - 8. The solution uses -2x^2 + 10x - 8 as the integrand, which is correct (top minus bottom). The bounds are 1 and 4. The result is 9. The solution is actually correct. I will mark it as pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The intersection points are incorrect. Solving -2x^2 + 7x - 8 = -3x yields -2x^2 + 10x - 8 = 0, which factors to -2(x-1)(x-4)=0, so the roots are indeed 1 and 4. Wait, let me re-check the problem statement vs the solution. The problem is y = -2x^2 + 7x - 8 and y = -3x. The difference is (-2x^2 + 7x - 8) - (-3x) = -2x^2 + 10x - 8. The roots of -2x^2 + 10x - 8 = 0 are x = 1 and x = 4. The integral of -2x^2 + 10x - 8 from 1 to 4 is indeed 9. The solution seems correct. Why did I think it was wrong? Let me re-read carefully. Ah, the problem says y = -2x^2 + 7x - 8. The solution uses -2x^2 + 10x - 8 as the integrand, which is correct (top minus bottom). The bounds are 1 and 4. The result is 9. The solution is actually correct. I will mark it as pass.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The setup in line 1 calculates the difference as -2x^2 + 10x - 8, which corresponds to subtracting y = 2x from the parabola, not y = -3x as stated in the problem. The correct difference should be -2x^2 + 10x - 8 if the line were y=2x, but for y=-3x, the difference is -2x^2 + 10x - 8? Wait. (-2x^2 + 7x - 8) - (-3x) = -2x^2 + 10x - 8. The algebra is actually correct for the subtraction. However, the roots of -2x^2 + 10x - 8 = 0 are x=1 and x=4. Let's check the intersection points. -2x^2 + 7x - 8 = -3x => -2x^2 + 10x - 8 = 0 => x^2 - 5x + 4 = 0 => (x-1)(x-4)=0. So the bounds are correct. The integral of -2x^2 + 10x - 8 from 1 to 4 is indeed 9. The solution is mathematically correct. Why did I think it was wrong? Let me re-read carefully. Problem: y = -2x^2 + 7x - 8 and y = -3x. Difference: (-2x^2 + 7x - 8) - (-3x) = -2x^2 + 10x - 8. Roots: 1, 4. Integral from 1 to 4 of (-2x^2 + 10x - 8) dx. Antiderivative: -2/3 x^3 + 5x^2 - 8x. Evaluated at 4: -128/3 + 80 - 32 = -128/3 + 48 = (-128 + 144)/3 = 16/3. Evaluated at 1: -2/3 + 5 - 8 = -2/3 - 3 = -11/3. Difference: 16/3 - (-11/3) = 27/3 = 9. The solution is correct. Wait, looking at line 1: `-2*(x - 4)*(x - 1), -2*x**2 + 10*x - 8`. This is the integrand. The sentence says 'Set the curves equal: top - bottom'. This is slightly imprecise phrasing (setting equal finds bounds, subtracting finds height), but the math follows. Is there a subtle error? The problem asks for area. The region is bounded. The parabola opens downward. The line is below the parabola between 1 and 4? Let's check x=2. Parabola: -2(4) + 14 - 8 = -8 + 6 = -2. Line: -3(2) = -6. -2 > -6. So parabola is on top. The setup is correct. The calculation is correct. The answer is correct. Why would this fail? Maybe the phrasing 'Set the curves equal: top - bottom' is confusing? It combines two steps. But it's not 'wrong'. Let me look closer at the prompt's constraints. 'Check that... the setup models the problem as stated'. It does. 'Each theorem... applied where hypotheses hold'. Yes. 'Nothing would teach a student something false'. It seems fine. Is there a trick? y = -3x. Maybe the user meant y = 3x? No, I must follow the problem statement. The problem statement says y = -3x. The solution uses y = -3x. The result is 9. It seems correct. I will mark it pass.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The setup in line 1 is incorrect: the difference between the curves is (-2x^2 + 7x - 8) - (-3x) = -2x^2 + 10x - 8, but the problem states the parabo
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.