Area between curves
Problem 5.118 · medium
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - x + 16 \) and \( \displaystyle y = 2 x - 2 \).
- \[ \left(6 - 3 x\right) \left(x + 3\right) = - 3 x^{2} - 3 x + 18 \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = -3 and x = 2; between them y = -3*x**2 - x + 16 is on top.
- \[ \int\limits_{-3}^{2} \left(- 3 x^{2} - 3 x + 18\right)\, dx = \frac{125}{2} \]Area = ∫ (top − bottom) dx.✓ Proved
Answer \( \frac{125}{2} \)
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.
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 - 3x + 18, but the problem states the top curve is -3x^2 - x + 16. The linear term in the difference should be -3x (from -x - 2x), not -3x, wait, -x - 2x = -3x. Let's re-read carefully. Top: -3x^2 - x + 16. Bottom: 2x - 2. Top - Bottom = -3x^2 - x - 2x + 16 - (-2) = -3x^2 - 3x + 18. The algebra in line 1 is actually correct for the difference. However, line 2 claims they cross at x=-3 and x=2. Let's check the roots of -3x^2 - 3x + 18 = 0. Divide by -3: x^2 + x - 6 = 0. (x+3)(x-2) = 0. Roots are -3 and 2. So the bounds are correct. The integral calculation is correct. Wait, did I miss something? Let's re-read the problem statement vs the solution. Problem: y = -3x^2 - x + 16. Solution line 1 says: -3*(x-2)*(x+3) which expands to -3(x^2+x-6) = -3x^2 - 3x + 18. This matches Top - Bottom. The roots are correct. The integral is correct. Why would it fail? Ah, look at line 2: "between them y = -3*x**2 - x + 16 is on top." This is true. Is there a subtle error? Let's check the area value. Integral of -3x^2 - 3x + 18 from -3 to 2. Antiderivative: -x^3 - 1.5x^2 + 18x. At 2: -8 - 6 + 36 = 22. At -3: -(-27) - 1.5(9) + 18(-3) = 27 - 13.5 - 54 = -40.5. 22 - (-40.5) = 62.5 = 125/2. The math is correct. Is the sentence misleading? No. Is the setup wrong? No. Wait, look at line 1 again. "-3*(x - 2)*(x + 3), -3*x**2 - 3*x + 18". The comment says "Set the curves equal: top - bottom, factored." The equation itself is just the factored form and expanded form of the difference. It is marked proved. Line 2 identifies the bounds. Line 3 computes the integral. Everything seems correct. Why would I fail it? Let me look closer at the prompt's instruction: "find what is wrong". Maybe there is nothing wrong? If there is nothing wrong, I should pass. But usually these adversarial reviews have a trick. Let's re-read the problem: y = -3x^2 - x + 16. Solution uses -3x^2 - 3x + 18 as the integrand. This is correct. Roots are -3, 2. Correct. Value 125/2. Correct. Is it possible the user made a typo in the problem statement provided to me? No, I must trust the problem statement. Is it possible the solution implies the curves are y = -3x^2 - 3x + 18 and y=0? No, it explicitly says "top - bottom". Okay, I will pass it.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The setup in line 1 is incorrect: the difference between the curves is -3x^2 - 3x + 18, but the problem states the top curve is -3x^2 - x + 16. The linear term in the difference should be -3x (from -x - 2x), not -3x, wait, -x - 2x = -3x. Let's re-read carefully. Top: -3x^2 - x + 16. Bottom: 2x - 2. Top - Bottom = -3x^2 - x - 2x + 16 - (-2) = -3x^2 - 3x + 18. The algebra in line 1 is actually correct for the difference. However, line 2 claims they cross at x=-3 and x=2. Let's check the roots of -3x^2 - 3x + 18 = 0. Divide by -3: x^2 + x - 6 = 0. (x+3)(x-2) = 0. Roots are -3 and 2. So the bounds are correct. The integral calculation is correct. Wait, did I miss something? Let's re-read the problem statement vs the solution. Problem: y = -3x^2 - x + 16. Solution line 1 says: -3*(x-2)*(x+3) which expands to -3(x^2+x-6) = -3x^2 - 3x + 18. This matches Top - Bottom. The roots are correct. The integral is correct. Why would it fail? Ah, look at line 2: "between them y = -3*x**2 - x + 16 is on top." This is true. Is there a subtle error? Let's check the area value. Integral of -3x^2 - 3x + 18 from -3 to 2. Antiderivative: -x^3 - 1.5x^2 + 18x. At 2: -8 - 6 + 36 = 22. At -3: -(-27) - 1.5(9) + 18(-3) = 27 - 13.5 - 54 = -40.5. 22 - (-40.5) = 62.5 = 125/2. The math is correct. Is the sentence misleading? No. Is the setup wrong? No. Wait, look at line 1 again. "-3*(x - 2)*(x + 3), -3*x**2 - 3*x + 18". The comment says "Set the curves equal: top - bottom, factored." The equation itself is just the factored form and expanded form of the difference. It is marked proved. Line 2 identifies the bounds. Line 3 computes the integral. Everything seems correct. Why would I fail it? Let me look closer at the prompt's instruction: "find what is wrong". Maybe there is nothing wrong? If there is nothing wrong, I should pass. But usually these adversarial reviews have a trick. Let's re-read the problem: y = -3x^2 - x + 16. Solution uses -3x^2 - 3x + 18 as the integrand. This is correct. Roots are -3, 2. Correct. Value 125/2. Correct. Is it possible the user made a typo in the problem statement provided to me? No, I must trust the problem statement. Is it possible the solution implies the curves are y = -3x^2 - 3x + 18 and y=0? No, it explicitly says "top - bottom". Okay, I will pass it.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-27 — The integrand in step 1 is incorrect; the difference between the curves is -3x^2 - 3x + 18, but the problem states y = -3x^2 - x + 16, so the correct difference is -3x^2 - 3x + 18 only if the constant term was 18, not 16. Wait, let's re-calculate: (-3x^2 - x + 16) - (2x - 2) = -3x^2 - 3x + 18. The algebra in step 1 is actually correct for the given functions. However, step 2 claims they cross at x = -3 and x = 2. Let's check: -3(-3)^2 - 3(-3) + 18 = -27 + 9 + 18 = 0. -3(2)^2 - 3(2) + 18 = -12 - 6 + 18 = 0. The roots are correct. The integral of -3x^2 - 3x + 18 from -3 to 2 is indeed 125/2. The solution is mathematically sound. Why did I think it was wrong? Let me re-read carefully. Problem: y = -3x^2 - x + 16. Line: y = 2x - 2. Difference: -3x^2 - 3x + 18. Roots of -3x^2 - 3x + 18 = 0 => x^2 + x - 6 = 0 => (x+3)(x-2)=0. Roots -3, 2. Integral is correct. The solution is correct. I will mark it as pass.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.