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Triple integrals in cylindrical coordinates

Problem 11.323 · easy

Use cylindrical coordinates to find \( \displaystyle \iiint_E x^{2} + y^{2}\, dV \) over the cylinder \( \displaystyle x^2 + y^2 \le 4 \), \( \displaystyle 0 \le z \le 1 \).

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.
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{2}\int\limits_{0}^{1} r^{3}\, dz\, dr\, d\theta = 8 \pi \]
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 sentence defines the radial bound as '0 <= r <= R' but fails to specify that R=2, which is required by the problem statement x^2 + y^2 <= 4. While the subsequent equation uses the correct bound, the setup sentence is incomplete and ambiguous.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The sentence defines the radial bound as '0 <= r <= R' but fails to specify that R=2, which is required by the problem statement x^2 + y^2 <= 4. While the subsequent equation uses the correct bound, the setup sentence is incomplete and ambiguous.
gpt-oss:20b: fail (error) 2026-10-09 — The first sentence omits the actual upper bound for r (it should be 2, not a generic R) and does not explicitly state the z‑bounds (0 to 1). This incomplete description could mislead a student about the limits of integration.
qwen3.6:27b-mlx: fail (style) 2026-10-09 — [domain objection, downgraded to style] The setup in line 1 is incomplete and ambiguous: it states 'z between the surfaces' without specifying them, and uses an undefined variable 'R' instead of the bound 2 derived from x^2 + y^2 <= 4. This fails to fully model the problem as stated.

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 2026-10-09 with SymPy 1.14.0.