Home›Calculus 3›Triple integrals in cylindrical coordinates›Problem 11.247

Triple integrals in cylindrical coordinates

Problem 11.247 · easy

Use cylindrical coordinates to find \( \displaystyle \iiint_E z\, dV \) over the cylinder \( \displaystyle x^2 + y^2 \le 4 \), \( \displaystyle 0 \le z \le 1 \).

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
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{2}\int\limits_{0}^{1} r z\, dz\, dr\, d\theta = 2 \pi \]
The iterated integral.✓ Proved

Answer \( 2 \pi \approx 6.28319 \)

Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

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 the cylinder (r from 0 to 2, theta from 0 to 2pi, z from 0 to 1) and the volume element. The calculation is correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The setup correctly identifies the bounds for the cylinder (r from 0 to 2, theta from 0 to 2pi, z from 0 to 1) and the volume element. The calculation is correct.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (style) 2026-10-05 — [domain objection, downgraded to style] The setup in step 1 uses an undefined variable 'R' for the radial bound instead of the specific value 2 derived from x^2 + y^2 <= 4. While step 2 uses the correct bound, the inconsistency in the setup description is an error.
gpt-oss:20b: fail (error) 2026-10-05 — The first sentence introduces the bounds incorrectly: it states 0 ≤ r ≤ R without specifying that R = 2, and it omits the explicit bounds 0 ≤ z ≤ 1. This incomplete description could mislead a student about the region’s limits.

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 2026-10-05 with SymPy 1.14.0.