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Volumes by disks and washers

Problem 5.93 · easy

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

Answer \( - \frac{\pi \left(1 - e^{4}\right)}{2} \)

✓ 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 applies the disk method for revolution about the x-axis. The setup is accurate, and the final answer is algebraically equivalent to the standard form (pi/2)(e^4 - 1).

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the disk method for revolution about the x-axis. The setup is accurate, and the final answer is algebraically equivalent to the standard form (pi/2)(e^4 - 1).
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The setup correctly identifies the method of disks for rotation about the x-axis, and the integral evaluation is correct.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The setup correctly identifies the method of disks for revolution about the x-axis, and the integral bounds and integrand are correct. The stated answer is algebraically equivalent to the standard form (e^4 - 1)pi/2.
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.