Triple integrals in cylindrical coordinates
Problem 11.203 · easy
Use cylindrical coordinates to find \( \displaystyle \iiint_E z\, dV \) over the cylinder \( \displaystyle x^2 + y^2 \le 4 \), \( \displaystyle 0 \le z \le 3 \).
- 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
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{2}\int\limits_{0}^{3} r z\, dz\, dr\, d\theta = 18 \pi \]The iterated integral.✓ Proved
Answer \( 18 \pi \approx 56.54867 \)
✓ 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: passqwen3.6:27b-mlx: pass — The setup correctly identifies the bounds for the cylinder and the integrand. The variable R in the text is implicitly resolved as 2 in the subsequent equation, which is consistent with x^2 + y^2 <= 4.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The setup correctly identifies the bounds for the cylinder and the integrand. The variable R in the text is implicitly resolved as 2 in the subsequent equation, which is consistent with x^2 + y^2 <= 4.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (style) 2026-10-04 — [domain objection, downgraded to style] The setup in line 1 uses an undefined variable 'R' for the radial bound instead of the specific value 2 derived from x^2 + y^2 <= 4. This makes the sentence mathematically incomplete and potentially confusing, even though the subsequent equation uses the correct bound.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.