Triple integrals in cylindrical coordinates
Problem 11.249 · easy
Use cylindrical coordinates to find \( \displaystyle \iiint_E z\, dV \) over the cylinder \( \displaystyle x^2 + y^2 \le 1 \), \( \displaystyle 0 \le z \le 1 \).
- 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.
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{1}\int\limits_{0}^{1} r z\, dz\, dr\, d\theta = \frac{\pi}{2} \]The iterated integral.✓ Proved
Answer \( \frac{\pi}{2} \approx 1.57080 \)
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 (style) — [domain objection, downgraded to style] The first sentence incorrectly specifies the bounds: it introduces an undefined variable R and omits the explicit limits for r (0 to 1) and z (0 to 1). The description of the region is therefore incomplete and misleading.qwen3.6:27b-mlx: fail (error) — The setup in line 1 is ambiguous and potentially incorrect because it uses a generic 'R' instead of the specific bound r=1 given in the problem. While the final answer is correct, the solution fails to explicitly state the bounds for r and z derived from the problem statement (0 <= r <= 1, 0 <= z <= 1) before jumping to the integral, making the connection between the problem and the calculation unclear.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The setup in line 1 is ambiguous and potentially incorrect because it uses a generic 'R' instead of the specific bound r=1 given in the problem. While the final answer is correct, the solution fails to explicitly state the bounds for r and z derived from the problem statement (0 <= r <= 1, 0 <= z <= 1) before jumping to the integral, making the connection between the problem and the calculation unclear.gpt-oss:20b: fail (style) 2026-10-05 — [domain objection, downgraded to style] The first sentence incorrectly specifies the bounds: it introduces an undefined variable R and omits the explicit limits for r (0 to 1) and z (0 to 1). The description of the region is therefore incomplete and misleading.qwen3.6:27b-mlx: fail (style) 2026-10-05 — [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. Line 2 correctly uses 1, but the text description is inconsistent and incomplete.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.