Question

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.

Answer 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

Answer 2

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