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Triple integrals in spherical coordinates

Problem 11.224 · medium

Use spherical coordinates to find the volume of the solid inside \( \displaystyle x^2 + y^2 + z^2 = 4 \) and above the cone \( \displaystyle z = \sqrt{x^2 + y^2} \).

ρ runs from 0 to 2, φ from 0 to pi/4, θ around the full circle; dV = ρ² sin φ dρ dφ dθ.
Reviewed
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{\frac{\pi}{4}}\int\limits_{0}^{2} \rho^{2} \sin{\left(\phi \right)}\, d\rho\, d\phi\, d\theta = \frac{8 \pi \left(2 - \sqrt{2}\right)}{3} \]
The iterated integral factors into three one-variable integrals.✓ Proved

Answer \( \frac{8 \pi \left(2 - \sqrt{2}\right)}{3} \approx 4.90747 \)

Lines: 1 proved, 1 reviewed. 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	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 over one symmetric piece

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer is incorrect; the correct volume is 8*pi*(2 - sqrt(2))/3, but the provided solution's final numerical approximation and the algebr
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/spherical_integral, checked 2026-10-05 with SymPy 1.14.0.