I'm trying to merge the ideas of single variable calculus and the hand-waving done in newtonian mechanics when differentials are used as a measure of infinitesimal quantity change $dy$ instead of a derivative function$\frac{dy}{dx}$, this quantities are then simplified and moved around as numbers. After a while this quantities are integrated to solve some diferential equation, for example $dy$ is integrated and you get $y(x)$ as a result.
$\int dy dx= y(x)$
I'm trying to formalize this concept a bit by using the definition of integral and derivative to try to get to the same results
Given the definition of a derivative
$$ \frac{dy}{dx} = \lim_{\Delta x\rightarrow0} \left( \frac{y(x+\Delta x)-y(x)}{\Delta x} \right)$$
I Could define
$$\frac{dy}{dx}dx = \lim_{\Delta x\rightarrow0} \left( \frac{y(x+\Delta x)-y(x)}{\Delta x} \Delta x \right) $$
And then define what looks like the idea used in physics, the change in the function (as opposed to the rate of change)
$$dy= \lim_{\Delta x\rightarrow0} \left( (y(x+\Delta x)-y(x)) \right) $$
I know that the area under a curve (the reinmann sum) is calculated as $$\int_a^b y(x) dx= \lim_{n\rightarrow \infty} \sum_{i=1}^n y\left((i\frac{b-a}{n})+a\right) \left(\frac{b-a}{n} \right)$$
If you where to integrate $dy$ it would look like this
to accomodate for the definition of the integral I make $\Delta x = \lim_{n\rightarrow \infty}\frac{b-a}{n}$
$$\int_a^b dy dx= \lim_{n\rightarrow \infty} \sum_{i=1}^n \left(y(x+\frac{b-a}{n})-y(x)\right) \left(\frac{b-a}{n} \right)$$
now this sum looks like an area to me, but it's kind of a funny area. let me make an abuse of notation to try to clarify what I'm reading here
$$\sum \left(y(x+\Delta x)-y(x)\right) \Delta x $$ $$\sum \left(\Delta y\right) \Delta x $$ This area kind of represents the sum of an infinite bunch of rectangles that follow the function plot between $x=a$ and $x=b$ and have $\Delta y$ height and $\Delta x$ width.
That's a funny result, I expected the sum of the instantaneous changes (the antiderivative).
what went wrong?
did any of this make any sense?