Question

find the volume of the portion of the solid sphere rholess than or equals8 that lies between the cones phiequalsStartFraction pi Over 6 EndFraction and phiequalsStartFraction 5 pi Over 6 EndFraction.

Answer 1

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