is $dx$ greater than $\frac{dx}{2}$? I wanted to ask if $dx$ is greater than $\frac{dx}{2}$? 
i will make conclusions i am sure they are wrong  : 
a) if yes then why in integration we do not use smaller than $dx$ like its half ? 
b) if you said they are equal then does it mean $1 = \frac{1}{2}$? some may say you can't divide by $dx$ but we do it a lot in solving differential equations ? so who is greater? might seem low question because  i am not math major like you
 A: To fully answer this question would require a many-volume narrative of the history of mathematics/physics since Newton and Leibniz! :)
But/and I would say that the question is eminently reasonable, rarely addressed directly in textbooks, and, indeed, subtle to answer "correctly".
As a ridiculously short sketch of what humans know about this, to the best of my own knowledge (and I am interested in such things for some years now):
Newton and Leibniz did argue/think genuinely in terms of "infinitesimals", and, yes, in that context, $dx/2$ is half as large as $dx$. (Yes, $dx$ is itself problemmatical in modern terms... though not at all impossibly so, in various ways, as "differential form", or as Skolem-Robinson-Nelson "infinitesimal").
Yes, tangential to foundational issues, differential equations can be solved by treating the various $d(whatever)$ as things existing in their own rights, without explaining what they are. That is, a heuristic succeeds in producing outcomes that are checkable.
The last 150 years of didactic tradition has been in a different direction, for somewhat artifactual reasons. That is, the popular style of calculus makes an exaggerated show of disparaging "infinitesimals" (despite Skolem-Robinson-Nelson's complete legitimization of them!), and of disparaging the symbol-manipulations that ... jeez! resolved zillions of questions over at least two centuries!
In short, the question is profoundly reasonable... but/and the accumulation of some centuries' artifacts about accepted mathematics does, indeed, seriously confuse anyone's understanding of ... for example... eminently reasonably heuristics in physics texts...
The operational answer is: try to think not in terms of "rules", but that the mathematics is mostly, and, certainly, initially, exactly a narrative, a description, of things. Then we hope that our subsequent manipulations of this description give us further information. 
That is, no, we cannot deduce by pure logic what the minimum legal parking distance away from a fire hydrant might be. But we can easily understand that there is some reasonable distance.
... sorry, yes, a seemingly vague answer, but, so far as I know, after some experience, maybe to the point.
A: In indefinite integral it is just a symbol. But it exist because indefinite and definite integral by Riman has very cool relationship as Newton-Leibniz theorem said.
In definite integral by Riman's definition it's small subarea in function domain that has infitine small diametr and take part in infinite summation. So in fact split your domain in which you're interesting as you wish for such dx-s...
BUT: that parts shouldn't overlap, and max(length(dx)) in limit should go to measure_of_length(point)=0....
Theorems which use integration are based only on such definition. 
So split as you want. 
Do define definitte classical integral you need to know classic definition of the limit. And define: function, function domain, how to measure area in function domain, what is a length, what is an operation of summation
Beside Riman's integral defintion, it is also exist Lebega integral which is used in theory of probability. As Software Engineer I know only that two integrals.
