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Area between curves

Problem 5.146 · medium

Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} - 4 x - 1 \) and \( \displaystyle y = 2 x + 3 \).

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

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

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.

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 returned a non-object

Every verdict on record (4)

qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer returned a non-object
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The setup in line 1 incorrectly factors the difference of the functions. The actual difference is (-2x^2 - 4x - 1) - (2x + 3) = -2x^2 - 6x - 4, but the factored form given is -2(x+1)(x+2) = -2x^2 - 6x - 4, which is algebraically correct for the integrand, BUT the roots of -2x^2 - 6x - 4 = 0 are x = -1 and x = -2. However, the problem states the curves are y = -2x^2 - 4x - 1 and y = 2x + 3. Let's re-verify the intersection points. -2x^2 - 4x - 1 = 2x + 3 => -2x^2 - 6x - 4 = 0 => x^2 + 3x + 2 = 0 => (x+1)(x+2) = 0. So x = -1, -2. The integrand is correct. The bounds are correct. The integral of -2x^2 - 6x - 4 from -2 to -1 is indeed 1/3. Wait, let me re-read line 1 carefully. It says '-2*(x + 1)*(x + 2), -2*x**2 - 6*x - 4'. This is just showing the factored form and expanded form of the integrand. It is not setting the curves equal to find roots, it is defining the integrand. Line 2 says they cross at -2 and -1. This is correct. Line 2 says y = -2x^2 - 4x - 1 is on top. Let's check x = -1.5. y1 = -2(2.25) - 4(-1.5) - 1 = -4.5 + 6 - 1 = 0.5. y2 = 2(-1.5) + 3 = 0. So y1 > y2. Correct. The integral calculation is correct. Why did I think it was wrong? Ah, I see. The prompt says 'Set the curves equal: top - bottom, factored.' in line 1. But line 1 is marked [equation, unchecked]. The text description in line 1 is 'Set the curves equal: top - bottom, factored.' This is a bit garbled but the math is right. Let me look closer at the problem statement vs solution. Problem: y = -2x^2 - 4x - 1 and y = 2x + 3. Solution integrand: -2x^2 - 6x - 4. This is ( -2x^2 - 4x - 1 ) - ( 2x + 3 ). Correct. Roots: -1, -2. Correct. Integral value: 1/3. Correct. Is there a subtle error? The solution says 'Set the curves equal' in the comment for line 1, but line 1 is just the integrand. This is a style issue or a misleading comment. But wait, look at line 1 again. '-2*(x + 1)*(x + 2), -2*x**2 - 6*x - 4'. This is not an equation. It's two expressions. The tag is [equation, unchecked]. Usually this implies LHS = RHS. If it's interpreted as an equation, it's nonsense. But the comment says 'Set the curves equal...'. This is confusing. However, the main issue might be simpler. Did the user make a typo in the problem statement or the solution? No. Let's look at the 'Stated answer: 1/3'. The solution arrives at 1/3. Everything seems correct. Why would it fail? Maybe the 'unchecked' tag on line 1 is problematic if it's not an equation? Or maybe the comment 'Set the curves equal' is misleading because line 1 is the integrand, not the equation for intersection? The intersection equation is -2x^2 - 6x - 4 = 0. Line 1 shows the expression. This is a 'style' or 'misleading' issue. But is it an error? The solution is mathematically sound. Let me re-read the instructions. 'Check that... each conclusion follows... nothing would teach a student something false.' The comment 'Set the curves equal' next to the integrand is misleading. Students might think setting curves equal gives the integrand directly without subtracting. But the integrand IS the difference. Setting them equal gives the roots. The comment conflates finding roots with defining the integrand. This is 'misleading'. Also, line 1 is marked [equation] but contains two expressions separated by a comma. This is not an equation. This is a formatting/style error in the solution representation. However, usually 'equation' lines are checked by CAS. If the CAS checked '-2*(x + 1)*(x + 2) == -2*x**2 - 6*x - 4', that's true. If it checked '-2*(x + 1)*(x + 2) = -2*x**2 - 6*x - 4' as an identity, it's true. The comment is the problem. It says 'Set the curves equal'. You don't set curves equal to get the area integrand; you subtract them. Setting them equal finds the bounds. This is a conceptual conflation in the text. Verdict: fail, severity: misleading.
gpt-oss:20b: pass 2026-09-28

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.