Volume of a Sphere using Spherical Coordinates
Spherical coordinates:
x = ρ sinφ cosθ
y = ρ sinφ sinθ
z = ρ cosφ
dV = ρ² sinφ dρ dφ dθ
Limits: 0 ≤ ρ ≤ R, 0 ≤ φ ≤ π, 0 ≤ θ ≤ 2π
V = ∭ dV = ∫₀^{2π} ∫₀^π ∫₀^R ρ² sinφ dρ dφ dθ
Integration steps:
> ρ first: [ρ³/3] from 0 to R → R³/3
> φ next: ∫ sinφ dφ from 0 to π = 2
> θ last: ∫ dθ from 0 to 2π = 2π
Final result: V = (4/3) π R³
This derivation is fundamental in physics, astronomy, fluid dynamics, and engineering for calculating volumes of spherical objects such as planets, droplets, cells, and particles.