Newton's Law of Universal Gravitation for extended objects Carroll and Ostlie write, in their book An Introduction to Modern Astrophysics (second edition), at the page 33, right after deriving Newton's Law of Universal Gravitation for two point masses using Kepler's third law and the three laws of motion :

Newton's law of gravity applies to any two objects having mass. In particular, for an extended object (as opposed to a point mass), the force exerted by that object on another extended object may be found by integrating over each of their mass distributions.

My question is : how is the integrating process a "particular" case of the formula for two point masses ? Shouldn't the integrating process really be a postulate on its own ?
 A: If you are really concerned about an axiomatic treatment, here is a method that takes the integral definition as a postulate, and then recovers the point mass situation as a special case: define the "interaction energy" of a pair of nonoverlapping mass distributions $\rho_1,\rho_2$ as 
$$U=G\int_{V_1}\int_{V_2}\frac{\rho_1(\mathbf{r}_1)\rho_2(\mathbf{r}_2)}{|\mathbf{r}_1-\mathbf{r}_2|}$$
where $V_1$ and $V_2$ are the relevant spatial extents of $\rho_1$ and $\rho_2$. While it's not immediately obvious that this definition is "correct" (although it does have the correct units of energy), it reproduces the experimentally observed behavior of gravitational interactions between objects. In particular, for the case of point masses $\rho_1(\mathbf{r})=m_1\delta(\mathbf{r}-\mathbf{a}),\rho_2(\mathbf{r})=m_2\delta(\mathbf{r}-\mathbf{b})$ one obtains (by integrating)
$$U=\frac{Gm_1m_2}{|\mathbf{a}-\mathbf{b}|}$$
which is exactly the potential form of Newton's law of gravitation for a pair of point masses (to get the more common force form, differentiate $U$ with respect to $(\mathbf{a},\mathbf{b})$).
