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What is a general formula for Earth gravity as a function of radius, given a spherical shell model?

I'm trying to derive the formula for gravitational acceleration as a function of Earth radius $g(r)$, given a spherical shell model where each shell has a constant density $\rho(r)$. I think it's something like:

$g(r) = \frac{4\pi G}{3r^{2}}[ \int_{0}^{r}\bar{\rho}(r)r^{3}.$d$r +F ]$

Where $F$ accounts for core mass. But I would like confirmation/references if possible! Thanks!