Calculating Mass of Spherical Planet: Solve Confusing Question

[jamesbob]

Question: A spehrical planet of Radius R has a density p which depends on the distance r from its centre according to the formula

$$p = \frac{p_0}{1 + (r/R)^2}$$​

where $$p_0$$ is a constant. By dividing the planet up into spherical shells of a small thickness dr, find the mass of the planet.

Ok so I am pretty confused on what I am to do here. Do i differentiate with respect to r?

 

[0rthodontist]

You want to find the mass of a spherical shell. Then you want to integrate that mass with r going from 0 to R to find the sum of all the spherical shells that make up the planet.

 

[HallsofIvy]

Imagine a shell of radius r, thickness dr. For small dr, its volume is approximately the surface area of the shell, $4\pi r^2$, times the thickness, dr: that is $dV= 4\pi r^2 dr$. Of course, the mass of that shell is the density at that radius times the volume:
$$4\pi\frac{p_0 r^2 dr}{1+\left(\frac{r}{R}\right)^2}$$
Integrate that from 0 to R.

 

[jamesbob]

ok thanks, and that will be the answer because by intergrating i find the sum of all the smaller parts?

 

[HallsofIvy]

That's a very rough way of putting it, but okay.