(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Using spherical coordinates, set up but DO NOT EVALUATE the triple integral of f(x,y,z) = x(x^2+y^2+z^2)^(-3/2) over the ball x^2 + y^2 + z^2 ≤ 16 where 2 ≤ z.

2. Relevant equations

x = ρ sin ϕ cos θ

y = ρ sin ϕ sin θ

z = ρ cos ϕ

ρ^2 = x^2 + y^2 + z^2

∫∫∫w f(x,y,z) dxdydz

= ∫from θ1 to θ2 ∫from ϕ1 to ϕ2 ∫from ρ1 to ρ2 f(ρ*sinϕcosθ,ρ*sinϕsinθ,ρ*cosϕ) (ρ^2*sinϕ dρ dϕ dθ)

3. The attempt at a solution

First, I graphed xy, xz, yz plane, and from there I tried to find θ, ϕ, and ρ. For ρ, I got 0≤ρ≤4, I am not too sure about the bottom bound (0). For ϕ, I got 0 ≤ ϕ≤ pi/4, I was told that the top bound (pi/4) is wrong. For θ, I got 0 ≤ θ ≤ pi, I think both the bounds are right for this one.

Then I plugged them into the triple intergal and changed x(x^2+y^2 + z^2)^(-3/2) into spherical coordinate to get "ρ^-2*sinϕcosθ"

∫from 0 to pi ∫from 0 to pi/4 ∫from 0 to 4 ρ^-2*sinϕcosθ (ρ^2*sinϕ dρ dϕ dθ)

= ∫from 0 to pi ∫from 0 to pi/4 ∫from 0 to 4 (sinϕ)^2 cosθ dρ dϕ dθ

Can you please help me find the correct bounds for θ, ϕ, and ρ and I believe the function in spherical coodinates is right. Thanks in advance.

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# Homework Help: Multivariable calculus, Integral using spherical coordinates

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