Find the volume of a region cut from solid sphere p=1 by the cone angle=(pi/3)

which principle applies

Use spherical coordinates:
V = ∫∫∫ 1 dV
...= ∫(θ = 0 to 2π) ∫(φ = 0 to π/3) ∫(ρ = 0 to 1) 1 * (ρ^2 sin φ dρ dφ dθ)
...= [∫(θ = 0 to 2π) dθ] * ∫(φ = 0 to π/3) sin φ dφ] * [∫(ρ = 0 to 1) ρ^2 dρ]
...= 2π * [-cos φ {for φ = 0 to π/3}] * [(1/3) ρ^3 {for ρ = 0 to 1}]
...= 2π * (1 - 1/2) * (1/3)
...= π/3.

I hope this helps!

The cone angle only controls φ.
The horizontal cross-sections are all full circles; hence θ = 0 to 2π.

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