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Triple integrals in cylindrical coordinates

Problem 11.295 · easy

Use cylindrical coordinates to find the volume of the solid inside \( \displaystyle x^2 + y^2 = 9 \), above \( \displaystyle z = 0 \) and below \( \displaystyle z = 12 - x^2 - y^2 \).

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.
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{3}\int\limits_{0}^{12 - r^{2}} r\, dz\, dr\, d\theta = \frac{135 \pi}{2} \]
The iterated integral.✓ Proved

Answer \( \frac{135 \pi}{2} \approx 212.05750 \)

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 sentence states "0 ≤ r ≤ R" but never defines R; it also omits the explicit bounds for z. This incomplete description could mislead a student about the correct limits of integration.
qwen3.6:27b-mlx: fail (error) — The setup sentence fails to specify the bound R=3, which is derived from the cylinder x^2 + y^2 = 9. Without explicitly stating r goes from 0 to 3, the student does not see how the integration limits in the next line are determined.

Every verdict on record (4)

gpt-oss:20b: fail (misleading) 2026-10-07 — The sentence states "0 ≤ r ≤ R" but never defines R; it also omits the explicit bounds for z. This incomplete description could mislead a student about the correct limits of integration.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The setup sentence fails to specify the bound R=3, which is derived from the cylinder x^2 + y^2 = 9. Without explicitly stating r goes from 0 to 3, the student does not see how the integration limits in the next line are determined.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to explicitly state the bounds for r and z in the text, leaving the setup incomplete. Specifically, it does not specify that r goes from 0 to 3 or that z goes from 0 to 12 - r^2, which are critical for the student to understand the region of integration.

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.