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Triple integrals in cylindrical coordinates

Problem 11.250 · easy

Use cylindrical coordinates to find the volume of the solid inside \( \displaystyle x^2 + y^2 = 1 \), above \( \displaystyle z = 0 \) and below \( \displaystyle z = 4 - x^2 - y^2 \).

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.
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{1}\int\limits_{0}^{4 - r^{2}} r\, dz\, dr\, d\theta = \frac{7 \pi}{2} \]
The iterated integral.✓ Proved

Answer \( \frac{7 \pi}{2} \approx 10.99557 \)

Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

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 (style) — [domain objection, downgraded to style] The solution fails to specify the value of R in the first sentence, leaving the radial bound undefined. Although the integral in line 2 uses r=1, the text does not explicitly state that R=1 is derived from the intersection of z=4-r^2 and z=0 (which actually occurs at r=2, not r=1; the cylinder x^2+y^2=1 imposes r=1, but the sentence structure implies R is a generic parameter without defining it).

Every verdict on record (4)

qwen3.6:27b-mlx: fail (style) 2026-10-05 — [domain objection, downgraded to style] The solution fails to specify the value of R in the first sentence, leaving the radial bound undefined. Although the integral in line 2 uses r=1, the text does not explicitly state that R=1 is derived from the intersection of z=4-r^2 and z=0 (which actually occurs at r=2, not r=1; the cylinder x^2+y^2=1 imposes r=1, but the sentence structure implies R is a generic parameter without defining it).
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05 — The setup correctly identifies the cylindrical coordinate transformation, the bounds for the region of integration (intersection of cylinder and paraboloid at r=1), and the volume element. The resulting integral and answer are correct.
gpt-oss:20b: pass 2026-10-05

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.