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Area between curves

Problem 5.62 · medium Video

Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 3 x + 15 \) and \( \displaystyle y = - x - 1 \).

\[ \left(8 - 2 x\right) \left(x + 2\right) = - 2 x^{2} + 4 x + 16 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -2 and x = 4; between them y = -2*x**2 + 3*x + 15 is on top.
\[ \int\limits_{-2}^{4} \left(- 2 x^{2} + 4 x + 16\right)\, dx = 72 \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( 72 \)

The curves y = -2x² + 3x + 15 and y = -x - 1 between x = -2 and x = 4. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 72.

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 + 3x + 15 = -x - 1 yields x = 3 and x = -2, not x = 4 and x = -2. Using the wrong bounds leads to an incorrect area.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The intersection points are incorrect. Solving -2x^2 + 3x + 15 = -x - 1 yields x = 3 and x = -2, not x = 4 and x = -2. Using the wrong bounds leads to an incorrect area.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The intersection points are incorrect. Solving -2x^2 + 3x + 15 = -x - 1 yields -2x^2 + 4x + 16 = 0, which simplifies to x^2 - 2x - 8 = 0, giving roots x = 4 and x = -2. Wait, let me re-check. (-2)(4)^2 + 3(4) + 15 = -32 + 12 + 15 = -5. -4 - 1 = -5. Correct. (-2)(-2)^2 + 3(-2) + 15 = -8 - 6 + 15 = 1. -(-2) - 1 = 1. Correct. The roots are indeed -2 and 4. The integral of -2x^2 + 4x + 16 from -2 to 4 is indeed 72. Why did I think it was wrong? Let me re-read the prompt. "Find the area of the region bounded by y = -2x^2 + 3x + 15 and y = -x - 1". The setup is correct. The calculation is correct. The answer is correct. Wait, is there a subtle error? Let's check the subtraction again. Top - Bottom = (-2x^2 + 3x + 15) - (-x - 1) = -2x^2 + 4x + 16. This matches line 1. The roots of -2x^2 + 4x + 16 = 0 are x = [-4 +/- sqrt(16 - 4(-2)(16))]/(-4) = [-4 +/- sqrt(16 + 128)]/(-4) = [-4 +/- sqrt(144)]/(-4) = [-4 +/- 12]/(-4). x1 = 8/-4 = -2. x2 = -16/-4 = 4. The bounds are correct. The integral is correct. The verdict should be pass. Why did I hesitate? Maybe I misread the problem statement? No. Maybe I misread the solution? No. Okay, it seems correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The intersection points are incorrect; solving -2x^2 + 3x + 15 = -x - 1 yields x = -2 and x = 3, not x = 4. Consequently, the integration bounds and the resulting area are wrong.
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.