Area between curves
Problem 5.172 · medium Video
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} + 11 x + 1 \) and \( \displaystyle y = 1 - x \).
- \[ - 3 x \left(x - 4\right) = - 3 x^{2} + 12 x \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = 0 and x = 4; between them y = -3*x**2 + 11*x + 1 is on top.
- \[ \int\limits_{0}^{4} \left(- 3 x^{2} + 12 x\right)\, dx = 32 \]Area = ∫ (top − bottom) dx.✓ Proved
Answer \( 32 \)
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 integrand in step 1 is incorrect; the difference between the curves is -3x^2 + 12x, but the problem states y = -3x^2 + 11x + 1, so the difference should be -3x^2 + 12x. Wait, let me re-calculate. Top: -3x^2 + 11x + 1. Bottom: 1 - x. Difference: (-3x^2 + 11x + 1) - (1 - x) = -3x^2 + 12x. The equation in step 1 says -3*x*(x - 4) = -3*x**2 + 12*x. This is algebraically correct. The roots are 0 and 4. The integral of -3x^2 + 12x from 0 to 4 is [-x^3 + 6x^2] from 0 to 4 = -64 + 96 = 32. The solution is actually correct. Why did I think it was wrong? Let me re-read carefully. Problem: y = -3x^2 + 11x + 1 and y = 1 - x. Difference: (-3x^2 + 11x + 1) - (1 - x) = -3x^2 + 12x. Roots of -3x^2 + 12x = 0 => -3x(x - 4) = 0 => x = 0, 4. Integral from 0 to 4 of (-3x^2 + 12x) dx = 32. The solution is correct. I will mark it as pass.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The integrand in step 1 is incorrect; the difference between the curves is -3x^2 + 12x, but the problem states y = -3x^2 + 11x + 1, so the difference should be -3x^2 + 12x. Wait, let me re-calculate. Top: -3x^2 + 11x + 1. Bottom: 1 - x. Difference: (-3x^2 + 11x + 1) - (1 - x) = -3x^2 + 12x. The equation in step 1 says -3*x*(x - 4) = -3*x**2 + 12*x. This is algebraically correct. The roots are 0 and 4. The integral of -3x^2 + 12x from 0 to 4 is [-x^3 + 6x^2] from 0 to 4 = -64 + 96 = 32. The solution is actually correct. Why did I think it was wrong? Let me re-read carefully. Problem: y = -3x^2 + 11x + 1 and y = 1 - x. Difference: (-3x^2 + 11x + 1) - (1 - x) = -3x^2 + 12x. Roots of -3x^2 + 12x = 0 => -3x(x - 4) = 0 => x = 0, 4. Integral from 0 to 4 of (-3x^2 + 12x) dx = 32. The solution is correct. I will mark it as pass.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-09-29 — 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 2026-09-29qwen3.6:27b-mlx: fail (error) 2026-09-29 — The setup in line 1 is incorrect: the difference between the curves is -3x^2 + 12x, but the problem states y = -3x^2 + 11x + 1, so the difference should be -3x^2 + 12x. Wait, let's re-calculate: (-3x^2 + 11x + 1) - (1 - x) = -3x^2 + 12x. The algebra in line 1 is actually correct. However, line 1 says 'Set the curves equal: top - bottom, factored'. Setting curves equal gives -3x^2 + 12x = 0, which factors to -3x(x-4)=0. This is correct. The integral in line 3 is correct. The answer 32 is correct. Why did I think it was wrong? Let me re-read carefully. Problem: y = -3x^2 + 11x + 1 and y = 1 - x. Difference: (-3x^2 + 11x + 1) - (1 - x) = -3x^2 + 12x. Roots: -3x(x-4) = 0 => x=0, x=4. Integral from 0 to 4 of (-3x^2 + 12x) dx = [-x^3 + 6x^2] from 0 to 4 = -(64) + 6(16) = -64 + 96 = 32. The solution is actually correct. The text in line 1 is slightly confusingly phrased ('Set the curves equal: top - bottom'), but the math follows. Is there a subtle error? Line 1: '-3*x*(x - 4), -3*x**2 + 12*x'. This represents the integrand. Line 2: 'They cross at x = 0 and x = 4; between them y = -3*x**2 + 11*x + 1 is on top.' This is true because the parabola opens downward and the line is below it between the roots. Line 3: Integral calculation is correct. Wait, look at line 1 again. It says 'Set the curves equal: top - bottom, factored.' This is a category error in description. You don't 'set the curves equal' to get the top-bottom difference. You subtract them. Setting them equal finds the intersection points. The label is misleading/wrong, but the math is right. The prompt asks to check if 'the setup models the problem as stated'. The setup *is* correct mathematically. The sentence in line 1 is a comment, not a sentence step? No, line 1 is marked [equation, unchecked]. The text after '--' is likely a comment. Let's look at line 2. It is a [sentence]. 'They cross at x = 0 and x = 4; between them y = -3*x**2 + 11*x + 1 is on top.' This is correct. Is there any error? Let's check the subtraction again. Top: -3x^2 + 11x + 1 Bottom: 1 - x Top - Bottom = -3x^2 + 11x + 1 - 1 + x = -3x^2 + 12x. Roots of -3x^2 + 12x = 0 are x=0 and x=4. Integral is 32. The solution seems perfectly correct. Why would it fail? Maybe the phrasing in line 1 'Set the curves equal: top - bottom' is considered 'misleading' because setting curves equal is for finding bounds, not for finding the integrand. But line 1 is an equation line. The comment is just a comment. Let's look closer at the prompt instructions. 'Check that... each conclusion follows from the lines before it'. Line 1 provides the integrand. Line 2 provides the bounds and confirms the top/bottom relationship. Line 3 computes the integral. It seems correct. I will mark it pass.gpt-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.