Triple integrals in cylindrical coordinates
Problem 11.248 · easy
Use cylindrical coordinates to find \( \displaystyle \iiint_E z\, dV \) over the cylinder \( \displaystyle x^2 + y^2 \le 9 \), \( \displaystyle 0 \le z \le 4 \).
- 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.Reviewed
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{3}\int\limits_{0}^{4} r z\, dz\, dr\, d\theta = 72 \pi \]The iterated integral.✓ Proved
Answer \( 72 \pi \approx 226.19467 \)
✓ 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 solution correctly sets up the cylindrical coordinates, including the Jacobian factor r, and identifies the correct bounds for the cylinder. The final result is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly sets up the cylindrical coordinates, including the Jacobian factor r, and identifies the correct bounds for the cylinder. The final result is correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-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 3 derived from x^2 + y^2 <= 9. Although the subsequent equation uses the correct bound, the sentence fails to establish the correct model parameters.gpt-oss:20b: pass 2026-10-05
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.