Mass of sphere with density ρ(r) = ρ0(1 − r/R)

[~Sam~]

Homework Statement

The density of a sphere of radius R as a function of distance from the centre r is ρ(r) = ρ₀(1 − r/R)
What is the mass of the sphere in terms of ρ₀ and R?

Homework Equations

Volume of a Sphere=(4/3)piR³

The Attempt at a Solution

I'm somewhat confused by the question. To my understanding, I can determine the mass of the sphere by integrating from 0 to radius R. I would consider a spherical shell a distance r from the center and of thickness dr. Thus I integrate from 0 to R and get something like 4pir²(1-r/R)ρ(r)dr, which is the mass of the sphere. But I'm not sure if that's what is needed. Any ideas?

 

[rl.bhat]

dm = 4pir2(1-r/R)ρ(r)dr
Complete the integration. So
m = 4*pi*rho* intg(r^2*dr - r^3*dr/R) from zero to R.

 

[~Sam~]

dm = 4pir2(1-r/R)ρ(r)dr
Complete the integration. So
m = 4*pi*rho* intg(r^2*dr - r^3*dr/R) from zero to R.

Huh, I'm not quite sure how the integration with (1-r/R) works...R would be the radius, and r would be the distance from centre...care to explain?

 

[rl.bhat]

You have to integrate [ intg(r^2*dr - r^3*dr/R)] between 0 to R.
It is equal to [R^3/3 - R^3/4]