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Area between curves

Problem 5.146 · medium Video

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

The curves y = -2x² - 4x - 1 and y = 2x + 3 between x = -2 and x = -1. 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: fail (error) — The setup in line 1 incorrectly defines the integrand as -2x^2 - 6x - 4, which is not the difference between the given curves (top - bottom). The actual difference is -2x^2 - 6x - 4 only if the bottom curve was 2x+3 and the top was -2x^2-4x-1, but the algebra in line 1 claims -2(x+1)(x+2) equals that quadratic, which is false (-2(x^2+3x+2) = -2x^2-6x-4). Wait, -2(x+1)(x+2) IS -2x^2-6x-4. Let's re-read carefully. The problem asks for area between y = -2x^2 - 4x - 1 and y = 2x + 3. The difference (top - bottom) is (-2x^2 - 4x - 1) - (2x + 3) = -2x^2 - 6x - 4. So the integrand is correct. However, line 1 says '-2*(x + 1)*(x + 2) = -2*x**2 - 6*x - 4'. This is algebraically correct. But where do the bounds come from? Setting -2x^2 - 6x - 4 = 0 gives -2(x^2 + 3x + 2) = 0 => x^2 + 3x + 2 = 0 => (x+1)(x+2)=0. So x = -1, -2. The bounds are correct. The integral of -2x^2 - 6x - 4 from -2 to -1 is indeed 1/3? Let's check. Antiderivative is -2/3 x^3 - 3x^2 - 4x. At -1: -2/3(-1) - 3(1) - 4(-1) = 2/3 - 3 + 4 = 2/3 + 1 = 5/3. At -2: -2/3(-8) - 3(4) - 4(-2) = 16/3 - 12 + 8 = 16/3 - 4 = 4/3. Difference: 5/3 - 4/3 = 1/3. The calculation is correct. Why did I think it was wrong? Ah, line 1 says 'Set the curves equal: top - bottom, factored.' It equates the factored form of the difference to the expanded form. This is just an algebraic identity check. The sentence in line 2 says 'They cross at x = -2 and x = -1'. This is derived from setting the difference to 0. This is correct. Is there a subtle error? 'between them y = -2*x**2 - 4*x - 1 is on top'. Let's check x = -1.5. Top: -2(2.25) - 4(-1.5) - 1 = -4.5 + 6 - 1 = 0.5. Bottom: 2(-1.5) + 3 = 0. So Top > Bottom. Correct. The solution seems correct. Wait, look at line 1 again. '-2*(x + 1)*(x + 2) = -2*x**2 - 6*x - 4'. This is marked [equation, proved]. It is an identity. It doesn't explicitly state that this expression IS the difference. The comment says 'Set the curves equal: top - bottom, factored.' This is a bit ambiguous. Does it mean 'Set the curves equal to find bounds' or 'The expression for top-bottom is this'? The comment implies the latter. The sentence in line 2 confirms the bounds and the top/bottom relationship. The integral in line 3 uses the correct integrand and bounds. The result is correct. Is there a 'misleading' aspect? The comment in line 1 is slightly confusingly phrased ('Set the curves equal' usually implies finding intersection points, but the equation shown is an expansion of the difference). However, the logic holds together. Let's look closer. Is the area 1/3? Yes. Is the setup correct? Yes. Is there any false teaching? No. Verdict: pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The setup in line 1 incorrectly defines the integrand as -2x^2 - 6x - 4, which is not the difference between the given curves (top - bottom). The actual difference is -2x^2 - 6x - 4 only if the bottom curve was 2x+3 and the top was -2x^2-4x-1, but the algebra in line 1 claims -2(x+1)(x+2) equals that quadratic, which is false (-2(x^2+3x+2) = -2x^2-6x-4). Wait, -2(x+1)(x+2) IS -2x^2-6x-4. Let's re-read carefully. The problem asks for area between y = -2x^2 - 4x - 1 and y = 2x + 3. The difference (top - bottom) is (-2x^2 - 4x - 1) - (2x + 3) = -2x^2 - 6x - 4. So the integrand is correct. However, line 1 says '-2*(x + 1)*(x + 2) = -2*x**2 - 6*x - 4'. This is algebraically correct. But where do the bounds come from? Setting -2x^2 - 6x - 4 = 0 gives -2(x^2 + 3x + 2) = 0 => x^2 + 3x + 2 = 0 => (x+1)(x+2)=0. So x = -1, -2. The bounds are correct. The integral of -2x^2 - 6x - 4 from -2 to -1 is indeed 1/3? Let's check. Antiderivative is -2/3 x^3 - 3x^2 - 4x. At -1: -2/3(-1) - 3(1) - 4(-1) = 2/3 - 3 + 4 = 2/3 + 1 = 5/3. At -2: -2/3(-8) - 3(4) - 4(-2) = 16/3 - 12 + 8 = 16/3 - 4 = 4/3. Difference: 5/3 - 4/3 = 1/3. The calculation is correct. Why did I think it was wrong? Ah, line 1 says 'Set the curves equal: top - bottom, factored.' It equates the factored form of the difference to the expanded form. This is just an algebraic identity check. The sentence in line 2 says 'They cross at x = -2 and x = -1'. This is derived from setting the difference to 0. This is correct. Is there a subtle error? 'between them y = -2*x**2 - 4*x - 1 is on top'. Let's check x = -1.5. Top: -2(2.25) - 4(-1.5) - 1 = -4.5 + 6 - 1 = 0.5. Bottom: 2(-1.5) + 3 = 0. So Top > Bottom. Correct. The solution seems correct. Wait, look at line 1 again. '-2*(x + 1)*(x + 2) = -2*x**2 - 6*x - 4'. This is marked [equation, proved]. It is an identity. It doesn't explicitly state that this expression IS the difference. The comment says 'Set the curves equal: top - bottom, factored.' This is a bit ambiguous. Does it mean 'Set the curves equal to find bounds' or 'The expression for top-bottom is this'? The comment implies the latter. The sentence in line 2 confirms the bounds and the top/bottom relationship. The integral in line 3 uses the correct integrand and bounds. The result is correct. Is there a 'misleading' aspect? The comment in line 1 is slightly confusingly phrased ('Set the curves equal' usually implies finding intersection points, but the equation shown is an expansion of the difference). However, the logic holds together. Let's look closer. Is the area 1/3? Yes. Is the setup correct? Yes. Is there any false teaching? No. Verdict: pass.
gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.