Area between curves
Problem 5.66 · medium Video
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 16 x - 11 \) and \( \displaystyle y = x + 1 \).
- \[ \left(12 - 3 x\right) \left(x - 1\right) = - 3 x^{2} + 15 x - 12 \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = 1 and x = 4; between them y = -3*x**2 + 16*x - 11 is on top.Reviewed
- \[ \int\limits_{1}^{4} \left(- 3 x^{2} + 15 x - 12\right)\, dx = \frac{27}{2} \]Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{27}{2} \)
Lines: 2 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the intersection points, determines the correct integrand (top curve minus bottom curve), and computes the definite integral accurately.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the intersection points, determines the correct integrand (top curve minus bottom curve), and computes the definite integral accurately.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the intersection points, determines the correct integrand (top curve minus bottom curve), and computes the definite integral accurately.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The setup in line 1 is incorrect: the difference between the curves is -3x^2 + 15x - 12, but the problem states the top curve is -3x^2 + 16x - 11 and the bottom is x + 1, so the difference should be -3x^2 + 15x - 12. Wait, let me re-calculate. (-3x^2 + 16x - 11) - (x + 1) = -3x^2 + 15x - 12. The equation in line 1 shows -3*(x-4)*(x-1) which expands to -3(x^2 - 5x + 4) = -3x^2 + 15x - 12. So the algebra is correct. However, line 2 claims they cross at x=1 and x=4. Let's check the roots of -3x^2 + 15x - 12 = 0. Dividing by -3: x^2 - 5x + 4 = 0 => (x-4)(x-1)=0. So x=1, 4. This is correct. The integral is from 1 to 4. The result 27/2 is correct. Why did I think it was wrong? Let me re-read carefully. Ah, the prompt says 'Find the area...'. The solution calculates the integral of the difference. Is the top curve actually on top? At x=2, y1 = -3(4) + 32 - 11 = -12 + 32 - 11 = 9. y2 = 2 + 1 = 3. 9 > 3. Yes. So the setup is correct. The algebra is correct. The bounds are correct. The result is correct. Wait, looking at line 1 again: '-3*(x - 4)*(x - 1), -3*x**2 + 15*x - 12'. The comment says 'Set the curves equal: top - bottom'. But the equation shown is just the factored form and the expanded form of the difference. It doesn't explicitly show the subtraction step, but it provides the correct integrand. Is there a subtle error? Let's check the intersection points again. -3x^2 + 16x - 11 = x + 1 => -3x^2 + 15x - 12 = 0. Correct. The area is indeed 27/2. The solution seems sound. Let me look for 'misleading' or 'style' issues. Line 1 is an equation marked proved. It equates two expressions. It's a bit weird to present the integrand as an equation to itself, but it's not wrong. Line 2 is a sentence. It correctly identifies bounds and relative position. Line 3 is the integral. Everything seems correct. Why would it fail? Maybe I missed something. Let's re-read the problem statement. y = -3x^2 + 16x - 11 and y = x + 1. Difference: -3x^2 + 15x - 12. Roots 1, 4. Integral from 1 to 4 of (-3x^2 + 15x - 12) dx. Antiderivative: -x^3 + 15/2 x^2 - 12x. Evaluated at 4: -64 + 15/2(16) - 48 = -64 + 120 - 48 = 8. Evaluated at 1: -1 + 7.5 - 12 = -5.5. 8 - (-5.5) = 13.5 = 27/2. The math is perfect. Is there a formatting issue? No. Is there a logical gap? No. I will mark it as pass.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.