Need help with a derivation/proof showing that gravity linearly decreases to zero from the surface to the center of a constant density planet. I'd like to do this by a triple integral of dV elements with a constant density using Newton's law of gravitation:
$$F_g=G\frac{m_1m_2}{r^2}$$
Where $G$ is the gravitational constant, $m_1$ is the mass of the object moving towards the center of the planet, $m_2$ is the mass of the $dV$ volume element in the integral, and $r$ is the distance from each dV element to $m_1$
With a substitution of $F_g=m_1a$, a simpler expression can perhaps be used independent of mass $m_1$ to solve for acceleration due to gravity, $a_g$
$$a_g=G\frac{m_2}{r^2}$$
Using either cylindrical or spherical coordinates, how would I best set up and solve this integral?