Volumes by cylindrical shells
Problem 5.126 · easy
The region under \( \displaystyle y = 12 - 3 x^{2} \) for \( \displaystyle 0 \le x \le 2 \) is revolved about the y-axis. Use shells to find the volume.
- A shell at x has radius x, height f(x) and thickness dx, so its volume is 2πx f(x) dx.Reviewed
- \[ \int\limits_{0}^{2} 2 \pi x \left(12 - 3 x^{2}\right)\, dx = 24 \pi \]V = ∫ 2πx f(x) dx.✓ Proved
Answer \( 24 \pi \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ 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 | 20,000 thin shells (annular slabs) summed numerically give the same volume |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The setup correctly identifies the radius, height, and thickness for the method of cylindrical shells. The integration bounds and integrand are correct, leading to the right answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The setup correctly identifies the radius, height, and thickness for the method of cylindrical shells. The integration bounds and integrand are correct, leading to the right answer.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The setup correctly identifies the radius, height, and bounds for the method of cylindrical shells. The integration is correct.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/volume_shells, checked 2026-09-27 with SymPy 1.14.0.