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Volumes by disks and washers

Problem 5.17 · easy

The region between \( \displaystyle y = x \) and \( \displaystyle y = x^{2} \) is revolved about the x-axis. Find the volume.

\[ - x \left(x - 1\right) = - x^{2} + x \]
The curves meet at x = 0 and x = 1.✓ Proved
Cross sections are washers: outer radius x, inner radius x**2.
Reviewed
\[ \int\limits_{0}^{1} \pi \left(- x^{4} + x^{2}\right)\, dx = \frac{2 \pi}{15} \]
V = ∫ A(x) dx.✓ Proved

Answer \( \frac{2 \pi}{15} \)

✓ Nihil obstat Lines: 2 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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 the cross-sectional area agrees

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the intersection points, the outer and inner radii for the washer method, and computes the integral accurately.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the intersection points, the outer and inner radii for the washer method, and computes the integral accurately.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the intersection points, the outer and inner radii for the washer method, and sets up the integral properly. The final result is correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The setup correctly identifies the intersection points, the outer and inner radii for the washer method, and the resulting integral evaluates to the correct volume.
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.