- #1
Bergen
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Homework Statement
A sphere is given by [itex]x^2+y^2+z^2 ≤ 1[/itex]. The density is given by [itex]ρ(x,y,z) = x+y+z[/itex].
Show that the mass is [itex]3π/2[/itex].
Homework Equations
[itex]m = ∫ρ ∂V[/itex]
[itex]∂V=ρ^2sinϕ∂ρ∂ϕ∂θ[/itex]
The Attempt at a Solution
I have converted the x, y and z in the density function to spherical coordinates, and by using triple integrals this is where I am now.
[itex]m = ∫∫∫(ρsin(ϕ)cos(θ)+ρsin(ϕ)sin(θ)+ρcos(ϕ))ρ^2sinθdρdθdϕ[/itex]
I am struggeling to solve the integration due to the density function in the bracket.
Integration limits; [itex]θ = 2π→0. ϕ = π→0, ρ = 1→0[/itex]
Best regards