Triple integrals in cylindrical coordinates
Problem 11.202 · easy
Use cylindrical coordinates to find \( \displaystyle \iiint_E z\, dV \) over the cylinder \( \displaystyle x^2 + y^2 \le 9 \), \( \displaystyle 0 \le z \le 3 \).
- 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.
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{3}\int\limits_{0}^{3} r z\, dz\, dr\, d\theta = \frac{81 \pi}{2} \]The iterated integral.✓ Proved
Answer \( \frac{81 \pi}{2} \approx 127.23450 \)
Lines: 1 proved, 1 not checked. 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 | 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: fail (misleading) — The first sentence claims the region is 0≤r≤R but never defines R; it should state r ranges from 0 to 3. This omission could mislead a student into thinking the radius is unspecified.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution uses an undefined variable 'R' in the sentence describing the bounds, failing to specify that R=3 based on the problem statement x^2 + y^2 <= 9. This makes the setup incomplete and ambiguous.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-04 — [domain objection, downgraded to style] The solution uses an undefined variable 'R' in the sentence describing the bounds, failing to specify that R=3 based on the problem statement x^2 + y^2 <= 9. This makes the setup incomplete and ambiguous.gpt-oss:20b: fail (misleading) 2026-10-04 — The first sentence claims the region is 0≤r≤R but never defines R; it should state r ranges from 0 to 3. This omission could mislead a student into thinking the radius is unspecified.qwen3.6:27b-mlx: fail (style) 2026-10-04 — [domain objection, downgraded to style] The solution uses an undefined variable 'R' in the sentence describing the bounds, failing to explicitly state that R=3 based on the problem statement x^2 + y^2 <= 9. Additionally, the equation line is marked 'unchecked', violating the constraint that equations are pre-verified by a CAS.gpt-oss:20b: fail (style) 2026-10-04 — [domain objection, downgraded to style] The setup sentence refers to an undefined variable R for the radial bound; it should explicitly state that r ranges from 0 to 3, matching the cylinder radius. This omission makes the description incomplete and potentially misleading.
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.