Answer:
V = (1024π√3)/3
Step-by-step explanation:
We can apply the formula
V = ∫∫∫ρ²SinФ dρ dФ d∅
1) ∫ρ²dρ = (1/3)ρ³ if 0 ≤ ρ ≤ 8
⇒ ∫ρ²dρ = (1/3)((8)³-(0)³) = 512/3
2) ∫ SinФ dФ = - Cos Ф if π/6 ≤ Ф ≤ 5π/6
⇒ ∫ SinФ dФ = - (Cos (5π/6) - Cos (π/6)) = √3
3) ∫ d∅ = ∅ if 0 ≤ ∅ ≤ 2π
⇒ ∫ d∅ = (2π - 0) = 2π
Finally, we get
V = ∫∫∫ρ²SinФ dρ dФ d∅ = (512/3)(√3)(2π) = (1024π√3)/3
