Calculating Pressure at the Center of the Earth

[derravaragh]

Homework Statement

Assume that the interior of the Earth is an incompressible fluid. The density is constant: ρ = M/V. The pressure p(r) depends on the distance r from the center of the earth. The equation for static equilibrium of a self-gravitating fluid sphere is
p(r)δA − p(r+dr)δA − ρdrδAg(r) = 0,
where g(r) = G (ρ4πr³/3) /r².

Homework Equations

The Attempt at a Solution

I need to solve this for p(r), so first I divided out the δA. From here I added terms to obtain:
p(r) = p(r+dr) + ρdrg(r)
Now I am stuck. I am unsure of how to deal with p(r+dr) and dr. My gut tells me to integrate with respect to r, but I don't feel like I should be integrating p(r). Other than that, when I look at p(r+dr) I see it as the pressure of the current r + the change in r, but that still doesn't help me proceed. Any help would be appreciated.

 

[mfb]

$$\lim_{dr \to 0}\frac{p(r+dr)-p(r)}{dr} = p'(r)$$
Just the regular definition of the derivative.