Volumes by cylindrical shells
Problem 5.108 · easy
The region under \( \displaystyle y = 2 x^{2} \) 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
- \[ \int\limits_{0}^{2} 4 \pi x^{3}\, dx = 16 \pi \]V = ∫ 2πx f(x) dx.✓ Proved
Answer \( 16 \pi \)
✓ 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 shell method. The integration and final result are correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The setup correctly identifies the radius, height, and thickness for the shell method. The integration and final result are correct.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The setup correctly identifies the radius, height, and thickness for the shell method, and the resulting integral is evaluated correctly.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/volume_shells, checked 2026-09-26 with SymPy 1.14.0.