Area between curves
Problem 5.124 · medium
Find the area of the region bounded by \( \displaystyle y = - 3 x^{2} - x + 25 \) and \( \displaystyle y = - x - 2 \).
- \[ \left(9 - 3 x\right) \left(x + 3\right) = 27 - 3 x^{2} \]Set the curves equal: top − bottom, factored.✓ Proved
- They cross at x = -3 and x = 3; between them y = -3*x**2 - x + 25 is on top.
- \[ \int\limits_{-3}^{3} \left(27 - 3 x^{2}\right)\, dx = 108 \]Area = ∫ (top − bottom) dx.✓ Proved
Answer \( 108 \)
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 intersection points are incorrect; solving -3x^2 - x + 25 = -x - 2 yields x = ±3, but substituting x=3 into the original equations gives y=10 and y=-5, which are not equal. The correct intersections are at x = ±3 only if the linear term cancels, but here -x cancels, leaving -3x^2 + 25 = -2 => 3x^2 = 27 => x = ±3. Wait, let me re-check. -3(3)^2 - 3 + 25 = -27 - 3 + 25 = -5. -3 - 2 = -5. They do intersect at x=3. Let me check x=-3. -3(-3)^2 - (-3) + 25 = -27 + 3 + 25 = 1. -(-3) - 2 = 3 - 2 = 1. They do intersect. So the bounds are correct. However, the integrand in step 1 is `27 - 3*x**2`. Let's check top - bottom. Top: -3x^2 - x + 25. Bottom: -x - 2. Difference: (-3x^2 - x + 25) - (-x - 2) = -3x^2 - x + 25 + x + 2 = -3x^2 + 27. This matches. The integral of 27 - 3x^2 from -3 to 3 is [27x - x^3] from -3 to 3 = (81 - 27) - (-81 + 27) = 54 - (-54) = 108. The solution is actually correct. Why did I doubt it? Ah, I misread the problem or my mental math. Let's look closer. Is there a subtle error? "They cross at x = -3 and x = 3". Correct. "Between them y = ... is on top". Correct. "Integral ... 108". Correct. The solution seems sound. Wait, looking at the provided solution text again. Step 1 says `27 - 3*x**2`. Step 3 integrates this. The result is 108. The stated answer is 108. Everything seems correct. Is there a style issue? No. Is there a misleading part? No. I will mark it as pass.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The intersection points are incorrect; solving -3x^2 - x + 25 = -x - 2 yields x = ±3, but substituting x=3 into the original equations gives y=10 and y=-5, which are not equal. The correct intersections are at x = ±3 only if the linear term cancels, but here -x cancels, leaving -3x^2 + 25 = -2 => 3x^2 = 27 => x = ±3. Wait, let me re-check. -3(3)^2 - 3 + 25 = -27 - 3 + 25 = -5. -3 - 2 = -5. They do intersect at x=3. Let me check x=-3. -3(-3)^2 - (-3) + 25 = -27 + 3 + 25 = 1. -(-3) - 2 = 3 - 2 = 1. They do intersect. So the bounds are correct. However, the integrand in step 1 is `27 - 3*x**2`. Let's check top - bottom. Top: -3x^2 - x + 25. Bottom: -x - 2. Difference: (-3x^2 - x + 25) - (-x - 2) = -3x^2 - x + 25 + x + 2 = -3x^2 + 27. This matches. The integral of 27 - 3x^2 from -3 to 3 is [27x - x^3] from -3 to 3 = (81 - 27) - (-81 + 27) = 54 - (-54) = 108. The solution is actually correct. Why did I doubt it? Ah, I misread the problem or my mental math. Let's look closer. Is there a subtle error? "They cross at x = -3 and x = 3". Correct. "Between them y = ... is on top". Correct. "Integral ... 108". Correct. The solution seems sound. Wait, looking at the provided solution text again. Step 1 says `27 - 3*x**2`. Step 3 integrates this. The result is 108. The stated answer is 108. Everything seems correct. Is there a style issue? No. Is there a misleading part? No. I will mark it as pass.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The intersection points are incorrect; solving -3x^2 - x + 25 = -x - 2 yields x = +/- 3, but the integrand used (27 - 3x^2) corresponds to y = -3x^2gpt-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.