Home›Calculus 1›Volumes by cylindrical shells›Problem 5.127

Volumes by cylindrical shells

Problem 5.127 · easy

The region under \( \displaystyle y = 2 e^{- x^{2}} \) for \( \displaystyle 0 \le x \le 1 \) 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
\[ \int\limits_{0}^{1} 4 \pi x e^{- x^{2}}\, dx = - \frac{2 \pi}{e} + 2 \pi \]
V = ∫ 2πx f(x) dx.✓ Proved

Answer \( - \frac{2 \pi}{e} + 2 \pi \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The setup correctly identifies the radius, height, and thickness for the shell method. The integration limits and integrand are correct, leading to the right result.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-27 — The setup correctly identifies the radius, height, and thickness for the shell method. The integration limits and integrand are correct, leading to the right result.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27 — The setup correctly identifies the radius, height, and thickness for the shell method. The integral bounds and integrand are correct, and the final evaluation matches the stated answer.
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.