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Triple integrals in cylindrical coordinates

Problem 11.321 · easy

Use cylindrical coordinates to find \( \displaystyle \iiint_E z\, dV \) over the cylinder \( \displaystyle x^2 + y^2 \le 4 \), \( \displaystyle 0 \le z \le 4 \).

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.
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{2}\int\limits_{0}^{4} r z\, dz\, dr\, d\theta = 32 \pi \]
The iterated integral.✓ Proved

Answer \( 32 \pi \approx 100.53096 \)

Lines: 1 proved, 1 not checked. 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	Not checked	—	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: fail (style) — [domain objection, downgraded to style] The setup in line 1 uses an undefined variable 'R' for the radial bound, whereas the problem specifies x^2 + y^2 <= 4, implying R=2. Line 2 correctly uses 2, but the sentence fails to define the specific bounds derived from the problem statement.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (style) 2026-10-09 — [domain objection, downgraded to style] The setup in line 1 uses an undefined variable 'R' for the radial bound, whereas the problem specifies x^2 + y^2 <= 4, implying R=2. Line 2 correctly uses 2, but the sentence fails to define the specific bounds derived from the problem statement.
gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The setup correctly identifies the bounds for the cylinder (r from 0 to 2, theta from 0 to 2pi, z from 0 to 4) and the volume element. The final result is 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 2026-10-09 with SymPy 1.14.0.