Suppose that $y$ is a function of a variable $x$ ( shortly : $y=f(x)$) ; in that case $ dy = f'(x) dx $ ( this is just the definition of a differential).
With this definition in hand, I try to integrate $dy$ from $x=a$ to $x=b$.
$\int_{a}^{b} dy$
$= \int_{a}^{b} f'(x)dx$
$= f(b)-f(a)$. ( By the Fundamental Theorem Of Calculus)
But maybe should I have treated $dx$ also as a differential : $dx= [x]'\Delta x = (1) \Delta x = \Delta x$. In that case, should not the answer be different?
$\int_{a}^{b} dy$
$= \int_{a}^{b} f'(x)dx$
$ = \int_{a}^{b} f'(x) . \Delta x$
$ = \Delta x \int_{a}^{b} f'(x)$
$ = (b-a) [ f(b)-f(a)]$.
So my question is: are we allowed to treat variables of integration symbols as differentials?