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Area between curves

Problem 5.68 · medium Video

Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} + 6 x + 9 \) and \( \displaystyle y = 2 x + 3 \).

\[ \left(6 - 2 x\right) \left(x + 1\right) = - 2 x^{2} + 4 x + 6 \]
Set the curves equal: top − bottom, factored.✓ Proved
They cross at x = -1 and x = 3; between them y = -2*x**2 + 6*x + 9 is on top.
\[ \int\limits_{-1}^{3} \left(- 2 x^{2} + 4 x + 6\right)\, dx = \frac{64}{3} \]
Area = ∫ (top − bottom) dx.✓ Proved

Answer \( \frac{64}{3} \)

The curves y = -2x² + 6x + 9 and y = 2x + 3 between x = -1 and x = 3. Thin rectangles fill the gap between them, getting narrower until they fill the region exactly; its area is 64/3.

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: fail (error) — The setup in line 1 is incorrect: it factors the difference of the curves as -2(x-3)(x+1), which equals -2x^2 + 4x + 6, but the actual difference (top - bottom) is -2x^2 + 4x + 6. Wait, let's re-calculate. Top: -2x^2 + 6x + 9. Bottom: 2x + 3. Difference: -2x^2 + 4x + 6. Factoring -2x^2 + 4x + 6: -2(x^2 - 2x - 3) = -2(x-3)(x+1). The roots are indeed -1 and 3. The integral of -2x^2 + 4x + 6 from -1 to 3 is 64/3. The solution is actually correct. Why did I think it was wrong? Let me re-read carefully. Line 1 says '-2*(x - 3)*(x + 1), -2*x**2 + 4*x + 6'. This is the factored form and expanded form of the integrand. Line 2 states the bounds and which is on top. Line 3 computes the integral. The logic is sound. The verdict should be pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The setup in line 1 is incorrect: it factors the difference of the curves as -2(x-3)(x+1), which equals -2x^2 + 4x + 6, but the actual difference (top - bottom) is -2x^2 + 4x + 6. Wait, let's re-calculate. Top: -2x^2 + 6x + 9. Bottom: 2x + 3. Difference: -2x^2 + 4x + 6. Factoring -2x^2 + 4x + 6: -2(x^2 - 2x - 3) = -2(x-3)(x+1). The roots are indeed -1 and 3. The integral of -2x^2 + 4x + 6 from -1 to 3 is 64/3. The solution is actually correct. Why did I think it was wrong? Let me re-read carefully. Line 1 says '-2*(x - 3)*(x + 1), -2*x**2 + 4*x + 6'. This is the factored form and expanded form of the integrand. Line 2 states the bounds and which is on top. Line 3 computes the integral. The logic is sound. The verdict should be pass.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The setup in step 1 is incorrect: the difference (top - bottom) is -2x^2 + 4x + 6, but the problem states the top curve is -2x^2 + 6x + 9. The difference should be (-2x^2 + 6x + 9) - (2x + 3) = -2x^2 + 4x + 6. Wait, let me re-calculate. (-2x^2 + 6x + 9) - (2x + 3) = -2x^2 + 4x + 6. The equation in step 1 says '-2*(x - 3)*(x + 1), -2*x**2 + 4*x + 6'. Let's check the roots of -2x^2 + 4x + 6. -2(x^2 - 2x - 3) = -2(x-3)(x+1). Roots are 3 and -1. This matches step 2. So the algebra in step 1 is actually correct for the difference. Let me re-read the problem statement carefully. y = -2x^2 + 6x + 9 and y = 2x + 3. Difference: -2x^2 + 6x + 9 - 2x - 3 = -2x^2 + 4x + 6. Step 1 shows the factored form and the expanded form of the difference. Step 2 identifies the intersection points correctly. Step 3 integrates the difference. The solution seems correct. Why did I think it was wrong? Let me double check the intersection points. -2x^2 + 6x + 9 = 2x + 3 => -2x^2 + 4x + 6 = 0 => x^2 - 2x - 3 = 0 => (x-3)(x+1)=0. x=3, x=-1. Correct. Integral of -2x^2 + 4x + 6 from -1 to 3. Antiderivative: -2/3 x^3 + 2x^2 + 6x. Evaluated at 3: -2/3(27) + 2(9) + 6(3) = -18 + 18 + 18 = 18. Evaluated at -1: -2/3(-1) + 2(1) + 6(-1) = 2/3 + 2 - 6 = 2/3 - 4 = -10/3. Result: 18 - (-10/3) = 54/3 + 10/3 = 64/3. The solution is correct. I will mark it as 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 (top - bottom) is -2x^2 + 4x + 6, but the problem states the top curve is -2x^2 + 6x + 9. The correc
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.