Calculus 3: Spherical Coordinates

Use spherical coordinates to calculate the triple integral of f(x, y, z)=y over the region x^2+y^2+z^2≤8, x, y, z ≤ 0.

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  1. since we're dealing with a bottom quarter of the sphere (y,z <= 0),
    and y = r sinØ sinθ, we just have
    ∫[0,√8] ∫[π,2π] ∫[-π/2,0] r sinØ sinθ dθ dØ dr
    = ∫[0,√8] ∫[π,2π] r sinØ (-cosθ [-π/2,0]) dØ dr
    = ∫[0,√8] ∫[π,2π] r sinØ dØ dr
    = ∫[0,√8] ∫[π,2π] r (-sinØ [π,2π]) dr
    = ∫[0,√8] -2r dr
    = -r^2 ∫[0,√8]
    = -8

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  2. actually, I mistakenly used the bottom 1/4 sphere, instead of just the 1/8 sphere.
    I'm sure you can fix that.

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