Triple integrals in cylindrical coordinates
Problem 11.245 · easy
Use cylindrical coordinates to find the volume of the solid inside \( \displaystyle x^2 + y^2 = 9 \), above \( \displaystyle z = 0 \) and below \( \displaystyle z = 11 - x^2 - y^2 \).
- 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
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{3}\int\limits_{0}^{11 - r^{2}} r\, dz\, dr\, d\theta = \frac{117 \pi}{2} \]The iterated integral.✓ Proved
Answer \( \frac{117 \pi}{2} \approx 183.78317 \)
✓ 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 r (0 to 3) based on the cylinder x^2 + y^2 = 9, and the integration limits for z and theta are correct. The final result matches the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The setup correctly identifies the bounds for r (0 to 3) based on the cylinder x^2 + y^2 = 9, and the integration limits for z and theta are correct. The final result matches the stated answer.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (style) 2026-10-05 — [domain objection, downgraded to style] The setup in line 1 is incomplete because it leaves the radial bound 'R' undefined, failing to explicitly state that R=3 derived from the cylinder x^2 + y^2 = 9. While the integral in line 2 uses the correct bound, the solution text does not connect the problem statement to the specific limit used.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.