Home›Calculus 3›Triple integrals in cylindrical coordinates›Problem 11.293

Triple integrals in cylindrical coordinates

Problem 11.293 · easy

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

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.
Reviewed
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{2}\int\limits_{0}^{2} r^{3}\, dz\, dr\, d\theta = 16 \pi \]
The iterated integral.✓ Proved

Answer \( 16 \pi \approx 50.26548 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly sets up the cylindrical coordinates, identifies the bounds for the cylinder, and computes the integral accurately.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly sets up the cylindrical coordinates, identifies the bounds for the cylinder, and computes the integral accurately.
gpt-oss:20b: pass 2026-10-07 — The setup is essentially correct; the sentence could be clearer by explicitly stating that R=2 and that z ranges from 0 to 2.
qwen3.6:27b-mlx: pass 2026-10-07 — The setup correctly identifies the integrand as r^3 (since x^2+y^2=r^2 and dV=r dz dr dtheta) and the bounds for the cylinder of radius 2 and height 2. The calculation 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.