Hello. Would really appreciate some help with this problem; the full solution or guidance.
Problem
The sphere x2+y2+z2 = a2 has a areal density which varies linearly with the distance from the xy-plane. Areal denisty σ = k*abs(z)/a, where k is a constant [mass/area]. Calculate the total mass.
My tries
I know I should use sphereical coordinates:
x = a * sin(φ) * cos(θ)
y = a * sin(φ) * sin(θ)
z = a * cos(φ)
I know that I should get a double integral that has dφdθ at the end. The thing is, I don't know what my limits will be, or what I will integrate. I know I can't integrate the areal density right of the bat.
Problem
The sphere x2+y2+z2 = a2 has a areal density which varies linearly with the distance from the xy-plane. Areal denisty σ = k*abs(z)/a, where k is a constant [mass/area]. Calculate the total mass.
My tries
I know I should use sphereical coordinates:
x = a * sin(φ) * cos(θ)
y = a * sin(φ) * sin(θ)
z = a * cos(φ)
I know that I should get a double integral that has dφdθ at the end. The thing is, I don't know what my limits will be, or what I will integrate. I know I can't integrate the areal density right of the bat.