Triple integrals in cylindrical coordinates
Problem 11.244 · easy
Use cylindrical coordinates to find \( \displaystyle \iiint_E x^{2} + y^{2}\, dV \) over the cylinder \( \displaystyle x^2 + y^2 \le 9 \), \( \displaystyle 0 \le z \le 4 \).
- 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
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{3}\int\limits_{0}^{4} r^{3}\, dz\, dr\, d\theta = 162 \pi \]The iterated integral.✓ Proved
Answer \( 162 \pi \approx 508.93801 \)
Lines: 1 proved, 1 reviewed. 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 | 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 solution correctly identifies the integrand transformation and volume element in cylindrical coordinates. The bounds for r, theta, and z are correctly derived from the problem statement, and the final result is accurate.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the integrand transformation and volume element in cylindrical coordinates. The bounds for r, theta, and z are correctly derived from the problem statement, and the final result is accurate.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution setup in step 1 is incomplete and ambiguous; it states 'z between the surfaces' but fails to explicitly define the bounds for z (0 to 4) in the text, relying on the next line to provide them. While the final calculation is correct, the sentence does not fully model the problem as stated by omitting the specific z-limits.gpt-oss:20b: inconclusive 2026-10-05 — reviewer returned a non-object
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.