Find the mass of the solid bounded by two concentrical spheres if the density is inversely proportional to the radius

Question

A body limited by two spherical and concentrical surfaces whose radius are $r$ and $R$ $(R>r)$.

How can I using integration find the mass of the solid between the surfaces if the density of the material is inversely proportional to the distance to the center of the spheres.

I am given that the answer $2\pi (R^2-r^2)$ but frankly dont know how to approach the problem.

Answer 1

Surface area of sphere = ${4\pi r^2}$

Take a small thickness of dr. So volume of this is ${4\pi r^2dr}$

As mass is density*volume, we have mass of this strip as ${\frac{1}{r}}{4\pi r^2dr} = {4\pi rdr}$

Now integrate it from r to R

${\int 4\pi rdr}$ = ${2\pi R^2 - 2\pi r^2}$ = ${2\pi (R^2 - r^2)}$

QED