Home›Calculus 1›Volumes by disks and washers›Problem 5.24

Volumes by disks and washers

Problem 5.24 · easy

The region under \( \displaystyle y = \sqrt{x} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 2 \) is revolved about the x-axis. Find the volume.

Cross sections perpendicular to the x-axis are disks of radius f(x), with area πf(x)².
Reviewed
\[ \int\limits_{0}^{2} \pi x\, dx = 2 \pi \]
V = ∫ A(x) dx.✓ Proved

Answer \( 2 \pi \)

✓ 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. How we verify

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	numerical quadrature of the cross-sectional area agrees

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the method (disk method) and sets up the integral with the correct radius squared (x) and bounds. The calculation is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the method (disk method) and sets up the integral with the correct radius squared (x) and bounds. The calculation is correct.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the method of disks, sets up the integral with the correct radius squared (x) and bounds (0 to 2), and computes the result accurately.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the method of disks, substitutes f(x)^2 = x into the area formula, and evaluates the integral 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_disk_washer, checked 2026-09-28 with SymPy 1.14.0.