Triple integrals in cylindrical coordinates
Problem 11.289 · easy
Use cylindrical coordinates to find \( \displaystyle \iiint_E x^{2} + y^{2}\, dV \) 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.
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{1}\int\limits_{0}^{2} r^{3}\, dz\, dr\, d\theta = \pi \]The iterated integral.✓ Proved
Answer \( \pi \approx 3.14159 \)
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: passqwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The setup in step 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 as stated.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (style) 2026-10-07 — [domain objection, downgraded to style] The setup in step 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 as stated.gpt-oss:20b: fail (error) 2026-10-07 — The setup sentence incorrectly uses a generic upper bound R instead of the specific radius 1 for the cylinder, and it does not explicitly state the z‑bounds (0 to 2). This misstates the region and would mislead a student about the correct limits.qwen3.6:27b-mlx: fail (style) 2026-10-07 — [domain objection, downgraded to style] The solution setup contains a logical error by defining the radial bound as '0 <= r <= R' without specifying that R=1 for this specific problem, leaving the variable R undefined in the text. Additionally, the integral setup in line 2 is technically incorrect because it integrates r^3 with respect to z first, but the integrand r^3 does not depend on z, so the z-integration should yield a factor of 2, which is implicitly handled but not explicitly shown in the transition from the general formula to the specific iterated integral.
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 2026-10-07 with SymPy 1.14.0.