Area between curves
Problem 5.144 · medium
Find the area of the region bounded by \( \displaystyle y = - 2 x^{2} - 3 x - 4 \) and \( \displaystyle y = 3 x \).
- \[ \left(- 2 x - 2\right) \left(x + 2\right) = - 2 x^{2} - 6 x - 4 \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = -2 and x = -1; between them y = -2*x**2 - 3*x - 4 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: passqwen3.6:27b-mlx: fail (error) — The setup is incorrect: the integrand used (-2x^2 - 6x - 4) does not match the difference of the given functions (top - bottom = -2x^2 - 6x - 4 is wrong, it should be -2x^2 - 6x - 4? Wait. Top is y=-2x^2-3x-4, bottom is y=3x. Difference is -2x^2 - 6x - 4. Let's re-read carefully. The problem states y = -2x^2 - 3x - 4. The solution line 1 says '-2*(x + 1)*(x + 2), -2*x**2 - 6*x - 4'. This implies the difference is -2x^2 - 6x - 4. But the difference is (-2x^2 - 3x - 4) - (3x) = -2x^2 - 6x - 4. So the algebra in line 1 is actually correct for the difference. However, line 2 claims the curves cross at x=-2 and x=-1. Let's check: -2x^2 - 3x - 4 = 3x => -2x^2 - 6x - 4 = 0 => x^2 + 3x + 2 = 0 => (x+1)(x+2)=0. Roots are -1, -2. This is correct. Line 2 says 'between them y = -2*x**2 - 3*x - 4 is on top'. Let's check x=-1.5. y1 = -2(2.25) - 3(-1.5) - 4 = -4.5 + 4.5 - 4 = -4. y2 = 3(-1.5) = -4.5. -4 > -4.5, so top is correct. The integral of (-2x^2 - 6x - 4) from -2 to -1 is indeed 1/3. Wait, why did I think it was wrong? Let me re-read the prompt's provided solution text. Line 1: '-2*(x + 1)*(x + 2), -2*x**2 - 6*x - 4'. This is the factorization of the difference. Line 3 integrates this difference. The result 1/3 is correct. Is there a subtle error? Ah, look at line 1 again. It says 'Set the curves equal: top − bottom, factored.' But the equation shown is just the expression. It's a bit garbled but the math holds. Wait, let me look closer at the problem statement vs the solution. Problem: y = -2x^2 - 3x - 4. Solution line 1 uses -2x^2 - 6x - 4 as the integrand. This is correct (Top - Bottom). The roots are correct. The area is correct. Why would this fail? Maybe the 'proved' tag on line 1 is misleading because it's not an equation? Or maybe the sentence in line 2 is technically correct but the phrasing 'They cross at...' implies the original curves, which they do. Let's look for a 'misleading' or 'style' issue. Actually, looking at line 1, it lists two expressions. The first is the factored form of the *difference*, the second is the expanded form. The comment says 'Set the curves equal'. Setting curves equal gives -2x^2 - 6x - 4 = 0. The line doesn't show '= 0'. It just shows the expression. This is a style issue, not an error. Is there an error? Let's re-calculate the integral. Int(-2x^2 - 6x - 4) dx = [-2/3 x^3 - 3x^2 - 4x]. From -2 to -1. 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. 5/3 - 4/3 = 1/3. The math is correct. The setup is correct. The conclusion is correct. I will mark it pass.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The setup is incorrect: the integrand used (-2x^2 - 6x - 4) does not match the difference of the given functions (top - bottom = -2x^2 - 6x - 4 is wrong, it should be -2x^2 - 6x - 4? Wait. Top is y=-2x^2-3x-4, bottom is y=3x. Difference is -2x^2 - 6x - 4. Let's re-read carefully. The problem states y = -2x^2 - 3x - 4. The solution line 1 says '-2*(x + 1)*(x + 2), -2*x**2 - 6*x - 4'. This implies the difference is -2x^2 - 6x - 4. But the difference is (-2x^2 - 3x - 4) - (3x) = -2x^2 - 6x - 4. So the algebra in line 1 is actually correct for the difference. However, line 2 claims the curves cross at x=-2 and x=-1. Let's check: -2x^2 - 3x - 4 = 3x => -2x^2 - 6x - 4 = 0 => x^2 + 3x + 2 = 0 => (x+1)(x+2)=0. Roots are -1, -2. This is correct. Line 2 says 'between them y = -2*x**2 - 3*x - 4 is on top'. Let's check x=-1.5. y1 = -2(2.25) - 3(-1.5) - 4 = -4.5 + 4.5 - 4 = -4. y2 = 3(-1.5) = -4.5. -4 > -4.5, so top is correct. The integral of (-2x^2 - 6x - 4) from -2 to -1 is indeed 1/3. Wait, why did I think it was wrong? Let me re-read the prompt's provided solution text. Line 1: '-2*(x + 1)*(x + 2), -2*x**2 - 6*x - 4'. This is the factorization of the difference. Line 3 integrates this difference. The result 1/3 is correct. Is there a subtle error? Ah, look at line 1 again. It says 'Set the curves equal: top − bottom, factored.' But the equation shown is just the expression. It's a bit garbled but the math holds. Wait, let me look closer at the problem statement vs the solution. Problem: y = -2x^2 - 3x - 4. Solution line 1 uses -2x^2 - 6x - 4 as the integrand. This is correct (Top - Bottom). The roots are correct. The area is correct. Why would this fail? Maybe the 'proved' tag on line 1 is misleading because it's not an equation? Or maybe the sentence in line 2 is technically correct but the phrasing 'They cross at...' implies the original curves, which they do. Let's look for a 'misleading' or 'style' issue. Actually, looking at line 1, it lists two expressions. The first is the factored form of the *difference*, the second is the expanded form. The comment says 'Set the curves equal'. Setting curves equal gives -2x^2 - 6x - 4 = 0. The line doesn't show '= 0'. It just shows the expression. This is a style issue, not an error. Is there an error? Let's re-calculate the integral. Int(-2x^2 - 6x - 4) dx = [-2/3 x^3 - 3x^2 - 4x]. From -2 to -1. 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. 5/3 - 4/3 = 1/3. The math is correct. The setup is correct. The conclusion is correct. I will mark it pass.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-28 — The setup incorrectly defines the integrand as just the top curve (-2x^2 - 3x - 4) instead of the difference (top - bottom). Additionally, the factored form in line 1 corresponds to -2x^2 - 6x - 4, which is not the top curve given in the problem statement.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.