Area between curves
Problem 5.166 · medium Video
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - x + 7 \) and \( \displaystyle y = 2 x + 1 \).
- \[ \left(3 - 3 x\right) \left(x + 2\right) = - 3 x^{2} - 3 x + 6 \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = -2 and x = 1; between them y = -3*x**2 - x + 7 is on top.
- \[ \int\limits_{-2}^{1} \left(- 3 x^{2} - 3 x + 6\right)\, dx = \frac{27}{2} \]Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{27}{2} \)
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: passqwen3.6:27b-mlx: fail (error) — The setup in line 1 is incorrect: the difference between the curves is (-3x^2 - x + 7) - (2x + 1) = -3x^2 - 3x + 6, but the factorization shown, -3*(x - 1)*(x + 2), expands to -3x^2 - 3x + 6, which matches the difference, not the original curve. However, the text claims 'Set the curves equal: top - bottom, factored', implying the equation -3x^2 - 3x + 6 = 0 is used to find intersections. The roots of -3x^2 - 3x + 6 = 0 are indeed x=1 and x=-2. So the intersection points are correct. The integral of the difference is correct. Wait, let me re-read carefully. Line 1 says '-3*(x - 1)*(x + 2) = -3*x**2 - 3*x + 6'. This is an identity. The comment says 'Set the curves equal: top - bottom, factored'. This is confusing phrasing but mathematically the difference is correct. The integral is correct. The result is correct. Is there a subtle error? The problem asks for the area bounded by y = -3x^2 - x + 7 and y = 2x + 1. The difference is -3x^2 - 3x + 6. The roots are 1 and -2. The integral from -2 to 1 of (-3x^2 - 3x + 6) is 27/2. The solution seems correct. Why would it fail? Let's check the factorization again. -3(x-1)(x+2) = -3(x^2 + x - 2) = -3x^2 - 3x + 6. Correct. The intersections are correct. The top/bottom is correct. The integral is correct. The answer is correct. Perhaps the 'error' is in the phrasing of line 1? 'Set the curves equal' usually implies finding roots of f(x)-g(x)=0. The line shows an expansion of the factored form of the difference. It's a bit disjointed but not wrong. Let's look closer. Is it possible the user made a typo in the problem statement vs the solution? No. Is it possible the 'proved' tag hides an error? No, the algebra is correct. Maybe the severity is 'style' because line 1 is confusingly labeled? Or maybe I'm missing something. Let's look at line 2. 'They cross at x = -2 and x = 1'. Correct. 'Between them y = ... is on top'. Correct. Line 3. Integral is correct. Result is correct. I will mark it as pass unless I find a specific flaw. Wait, look at line 1 again. '-3*(x - 1)*(x + 2) = -3*x**2 - 3*x + 6'. The comment says '-- Set the curves equal: top − bottom, factored.' This implies that the expression on the left IS the top minus bottom. Top - Bottom = (-3x^2 - x + 7) - (2x + 1) = -3x^2 - 3x + 6. The factored form is -3(x-1)(x+2). So the equation is correct. The solution is sound.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The setup in line 1 is incorrect: the difference between the curves is (-3x^2 - x + 7) - (2x + 1) = -3x^2 - 3x + 6, but the factorization shown, -3*(x - 1)*(x + 2), expands to -3x^2 - 3x + 6, which matches the difference, not the original curve. However, the text claims 'Set the curves equal: top - bottom, factored', implying the equation -3x^2 - 3x + 6 = 0 is used to find intersections. The roots of -3x^2 - 3x + 6 = 0 are indeed x=1 and x=-2. So the intersection points are correct. The integral of the difference is correct. Wait, let me re-read carefully. Line 1 says '-3*(x - 1)*(x + 2) = -3*x**2 - 3*x + 6'. This is an identity. The comment says 'Set the curves equal: top - bottom, factored'. This is confusing phrasing but mathematically the difference is correct. The integral is correct. The result is correct. Is there a subtle error? The problem asks for the area bounded by y = -3x^2 - x + 7 and y = 2x + 1. The difference is -3x^2 - 3x + 6. The roots are 1 and -2. The integral from -2 to 1 of (-3x^2 - 3x + 6) is 27/2. The solution seems correct. Why would it fail? Let's check the factorization again. -3(x-1)(x+2) = -3(x^2 + x - 2) = -3x^2 - 3x + 6. Correct. The intersections are correct. The top/bottom is correct. The integral is correct. The answer is correct. Perhaps the 'error' is in the phrasing of line 1? 'Set the curves equal' usually implies finding roots of f(x)-g(x)=0. The line shows an expansion of the factored form of the difference. It's a bit disjointed but not wrong. Let's look closer. Is it possible the user made a typo in the problem statement vs the solution? No. Is it possible the 'proved' tag hides an error? No, the algebra is correct. Maybe the severity is 'style' because line 1 is confusingly labeled? Or maybe I'm missing something. Let's look at line 2. 'They cross at x = -2 and x = 1'. Correct. 'Between them y = ... is on top'. Correct. Line 3. Integral is correct. Result is correct. I will mark it as pass unless I find a specific flaw. Wait, look at line 1 again. '-3*(x - 1)*(x + 2) = -3*x**2 - 3*x + 6'. The comment says '-- Set the curves equal: top − bottom, factored.' This implies that the expression on the left IS the top minus bottom. Top - Bottom = (-3x^2 - x + 7) - (2x + 1) = -3x^2 - 3x + 6. The factored form is -3(x-1)(x+2). So the equation is correct. The solution is sound.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-09-29 — The intersection points are incorrect; solving -3x^2 - x + 7 = 2x + 1 yields x = 1 and x = -8/3, not x = -2. Consequently, the integrand and bounds are wrong.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: inconclusive 2026-09-29 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The setup in line 1 incorrectly factors the difference of the functions. The actual difference is -3x^2 - 3x + 6, but the problem states the curvesgpt-oss:20b: pass 2026-09-29
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 2026-10-04 with SymPy 1.14.0.