Triple integrals in spherical coordinates
Problem 11.274 · easy
Use spherical coordinates to find \( \displaystyle \iiint_E \frac{dV}{\sqrt{x^2 + y^2 + z^2}} \) over the ball of radius 1.
- ρ runs from 0 to 1, φ from 0 to pi, θ around the full circle; dV = ρ² sin φ dρ dφ dθ.Reviewed
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{\pi}\int\limits_{0}^{1} \rho \sin{\left(\phi \right)}\, d\rho\, d\phi\, d\theta = 2 \pi \]The iterated integral factors into three one-variable integrals.✓ Proved
Answer \( 2 \pi \approx 6.28319 \)
✓ 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 over one symmetric piece |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The setup correctly identifies the bounds for the unit ball and the Jacobian for spherical coordinates. The integrand simplification and the resulting iterated integral are correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The setup correctly identifies the bounds for the unit ball and the Jacobian for spherical coordinates. The integrand simplification and the resulting iterated integral are correct.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The setup correctly identifies the bounds for the unit ball and the integrand simplification (1/rho * rho^2 sin(phi) = rho sin(phi)). The factorization and final result are 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/spherical_integral, checked 2026-10-06 with SymPy 1.14.0.