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Triple integrals in cylindrical coordinates

Problem 11.298 · easy

Use cylindrical coordinates to find the volume of the solid inside \( \displaystyle x^2 + y^2 = 1 \), above \( \displaystyle z = 0 \) and below \( \displaystyle z = 3 - x^2 - y^2 \).

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.
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{1}\int\limits_{0}^{3 - r^{2}} r\, dz\, dr\, d\theta = \frac{5 \pi}{2} \]
The iterated integral.✓ Proved

Answer \( \frac{5 \pi}{2} \approx 7.85398 \)

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
qwen3.6:27b-mlx: fail (error) — The sentence defines the radial bound as '0 <= r <= R' but fails to specify that R=1, which is required by the cylinder x^2 + y^2 = 1. While the subsequent integral uses the correct limit, the setup sentence is incomplete and ambiguous.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The sentence defines the radial bound as '0 <= r <= R' but fails to specify that R=1, which is required by the cylinder x^2 + y^2 = 1. While the subsequent integral uses the correct limit, the setup sentence is incomplete and ambiguous.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the bounds for cylindrical coordinates and sets up the integral properly. The final result is correct.

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.