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17 July, 10:50

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.

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  1. 17 July, 11:17
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    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
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