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Triple integrals in cylindrical coordinates

Problem 11.200 · easy

Use cylindrical coordinates to find the volume of the solid inside \( \displaystyle x^2 + y^2 = 4 \), above \( \displaystyle z = 0 \) and below \( \displaystyle z = 4 - x^2 - y^2 \).

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.
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{2}\int\limits_{0}^{4 - r^{2}} r\, dz\, dr\, d\theta = 8 \pi \]
The iterated integral.✓ Proved

Answer \( 8 \pi \approx 25.13274 \)

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 solution fails to explicitly state the upper bound for r (R=2) in the sentence, which is derived from the intersection of the cylinder and the paraboloid. While the integral uses the correct limit, the setup sentence is incomplete and does not fully model the problem constraints.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly state the upper bound for r (R=2) in the sentence, which is derived from the intersection of the cylinder and the paraboloid. While the integral uses the correct limit, the setup sentence is incomplete and does not fully model the problem constraints.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly state the bounds for r and theta in the text, relying solely on the unchecked equation line. While the equation is correct, the sentence describing the setup is incomplete and does not fully model the problem as requested.
gpt-oss:20b: pass 2026-10-04

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 2026-10-04 with SymPy 1.14.0.