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Triple integrals in cylindrical coordinates

Problem 11.366 · easy

Use cylindrical coordinates to find the volume of the solid inside \( \displaystyle x^2 + y^2 = 4 \), above \( \displaystyle z = 0 \) and below \( \displaystyle z = 7 - x^2 - y^2 \).

In cylindrical coordinates x² + y² = r² and dV = r dz dr dθ; the region is 0 ≤ θ ≤ 2π, 0 ≤ r ≤ R and z between the surfaces.
Reviewed
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{2}\int\limits_{0}^{7 - r^{2}} r\, dz\, dr\, d\theta = 20 \pi \]
The iterated integral.✓ Proved

Answer \( 20 \pi \approx 62.83185 \)

✓ 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	nested quadrature in rectangular coordinates

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The setup correctly identifies the bounds for r (0 to 2) and theta (0 to 2pi) based on the cylinder x^2 + y^2 = 4. The z-bounds are correctly identified as 0 to 7 - r^2. The integral calculation is correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: pass 2026-10-12 — The setup correctly identifies the bounds for r (0 to 2) and theta (0 to 2pi) based on the cylinder x^2 + y^2 = 4. The z-bounds are correctly identified as 0 to 7 - r^2. The integral calculation is correct.
gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: pass 2026-10-12 — The setup correctly identifies the cylindrical coordinate transformations, the bounds for r (0 to 2 based on x^2+y^2=4), theta (0 to 2pi), and z (0 to 7-r^2). The resulting integral and answer are correct.

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/cylindrical_integral, checked 2026-10-12 with SymPy 1.14.0.