Triple integrals in cylindrical coordinates
Problem 11.324 · easy
Use cylindrical coordinates to find \( \displaystyle \iiint_E z\, dV \) over the cylinder \( \displaystyle x^2 + y^2 \le 1 \), \( \displaystyle 0 \le z \le 2 \).
- 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.
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{1}\int\limits_{0}^{2} r z\, dz\, dr\, d\theta = 2 \pi \]The iterated integral.✓ Proved
Answer \( 2 \pi \approx 6.28319 \)
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: pass — The setup is essentially correct, though the sentence could be clearer by explicitly stating r ≤ 1 and z between 0 and 2 instead of using an undefined R.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The setup in line 1 uses an undefined variable 'R' for the radial bound, whereas the problem specifies x^2 + y^2 <= 1, implying R=1. This makes the setup incomplete and technically incorrect until R is defined.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09 — The setup is essentially correct, though the sentence could be clearer by explicitly stating r ≤ 1 and z between 0 and 2 instead of using an undefined R.qwen3.6:27b-mlx: fail (style) 2026-10-09 — [domain objection, downgraded to style] The setup in line 1 uses an undefined variable 'R' for the radial bound, whereas the problem specifies x^2 + y^2 <= 1, implying R=1. This makes the setup incomplete and technically incorrect until R is defined.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — The setup in line 1 is ambiguous and potentially misleading because it defines the radial bound as '0 <= r <= R' without explicitly stating R=1, relying on the reader to infer it from the problem statement rather than the solution text. Furthermore, line 2 is marked 'unchecked' despite the prompt stating that equations are checked by a CAS; if it is unchecked, the verification of the integral evaluation is missing.
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 2026-10-09 with SymPy 1.14.0.