Triple integrals in spherical coordinates
Problem 11.277 · easy
Use spherical coordinates to find \( \displaystyle \iiint_E (x^2 + y^2 + z^2)\, dV \) over the ball \( \displaystyle x^2 + y^2 + z^2 \le 9 \).
- ρ runs from 0 to 3, φ from 0 to pi, θ around the full circle; dV = ρ² sin φ dρ dφ dθ.Reviewed
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{\pi}\int\limits_{0}^{3} \rho^{4} \sin{\left(\phi \right)}\, d\rho\, d\phi\, d\theta = \frac{972 \pi}{5} \]The iterated integral factors into three one-variable integrals.✓ Proved
Answer \( \frac{972 \pi}{5} \approx 610.72561 \)
✓ 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
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The setup correctly identifies the bounds for the ball of radius 3 and the integrand in spherical coordinates. 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.