This is going to be a rather general question about suggestions on best supplementary material to properly explain the use of infinitesimals (or differentials?) for the purposes of integration or differential equations. Sorry about the question being long.
I have been trying to self study AP physics for some time now. As a background, it might be helpful to know that I have some familiarity with (semi-rigorous) Calculus (up to differential equations). In the course of my study, I have found, many times, explanations of certain concepts using infinitesimals as its argument. I have found this difficult to understand (and difficult to replicate in practice questions) in a rigorous way because my study of calculus did not include much on the operations on infinitesimals (except for Taylor Series). I have always used $\epsilon,\delta$, supremums, infimums, $\forall$ statements, least-upper-bound axioms, ... to work with limits and integrals and so on. My calculus book took extreme care to revert to the original analytical definitions every time an new integral is set up for a physical problem and proved that doing it that way is mathematically correct.
But, many things in the book I'm using (Fundamentals of Physics I by R. Shankar) depended on infinitesimals (or differentials?) to solve certain differential equations, to integrate over a set of infinitesimally small quantities, or to explain momentary changes (e.g. rocket equations, gyroscopic precession, moment of inertia of hollow spheres (which is a practice question I'm currently stuck on)). I would greatly appreciate it if someone can suggest some supplementary material to explain the algebra of infinitesimals, why and in what context can we drop higher order infinitestimals, when can we directly integrate an equation of infinitesimals when we arrive at it (for example in the case of the rocket equation, after applying momentum principles we should get $\frac{-dm}{M}=\frac{dv}{v_0}$ where m is mass of the rocket as a function of t and v is velocity of the rocket as a function of t. Why can we integrate both sides?), etc.
To illustrate my point, take the case of the moment of inertia of hollow and uniformly dense spheres. Using the logic of infinitesimals (the way in which I currently understand it), I multiplied the mass per unit of surface area of the sphere by the surface area of the infinitesimally thick ring ($2\pi r dr$) times the appropriate factor to get the moment of inertia of a ring. I arrived at the following integral: $$I_S = \int_{-R}^{R}\frac{M}{4\pi R^2}(2\pi\sqrt{R^2-y^2})(R^2-y^2)dy$$$$I_S = \int_{-R}^{R}\frac{M}{4\pi R^2}(2\pi\sqrt{R^2-y^2})(R^2-y^2)dy.$$ After doing a trig sub, this reduces to $$\frac{MR^2}{2}\int_{-\pi/2}^{\pi/2}\cos^4(\theta)d\theta=\frac{3\pi}{16}MR^2$$$$\frac{MR^2}{2}\int_{-\pi/2}^{\pi/2}\cos^4(\theta)d\theta=\frac{3\pi}{16}MR^2.$$I know this is wrong and I sense somewhat that relative to the surface area the difference between an arc and a straight line parallel to axis is probably not trivial. But I would just like take this example highlight a deficiency in my calculus training (that did not systematically explain what operations on infinitesimals is allowable) that I hope to remedy. I did this because I've seen other people do similar things and I couldn't find a place where this is formally explained.
Thank you all very much for your help in advance. This bit is really getting me stuck sometimes.
