Definite integral $\int ^b _a f(x)dx = \lim_{n \to \infty} \sum^n _{k=1} f(\zeta_k) \Delta x_k$ This is the mathematical description of the definite integral of $f(x)$ between $a$ and $b$:
$$\int ^b _a f(x)dx =  \lim_{n \to \infty} \sum^n _{k=1}  f(\zeta_k) \Delta x_k$$
In here, $\Delta x_k$ is the width of a rectangle, $f(\zeta_k)$ is the height of a rectangle, $\displaystyle \lim_{n \to \infty} \displaystyle \sum^n _{k=1}$ just is the sum of all of the n number of rectangles (which approaches $\infty$), $\displaystyle \int ^b _a f(x)$ just means the integral of the function f(x) from point a to b. What is left over in this expression which I cannot really give its use here, is the $dx$. What does it stand for here? I know it comes from the $\dfrac{dy}{dx}$ but I still don't really understand it here.  
 A: The symbol "$dx$" serves as a label to indicate which variable is being integrated over. 
It is perfectly fine to leave out "$dx$" if no confusion can arise. But consider the case where $f$ is a multivariate function: then the label is essential.
Moreover, we can interpret $dx$ as a kind of limit: The definition of the (Riemann) integral is a limit of the sum over the partition of $[a,b]$ as each $\Delta x_k$ (uniformly) goes to zero. This limit is independent of how the limit is taken, as long as each $\Delta x_k$ goes uniormly to zero. Thus, in the limit we have an "infinite sum" over "infinitesimally small intervals of lenth $dx$".
In abstract integraton theory, one may integrate over wildly different kinds of objects. Here,   symbols like $dx$ stands for the abstract measure being used. It can, for example, include a weight function, such as Stieltjes integrals.
A: The $d$ notation indicates the variable of integration. You can see similar roles for the variable $k$ of summation in
$$\sum_{k = 1}^n a_k$$
and the variable $x$ of integration in 
$$\int_a^b f(x)\, dx.$$
Both variables are "dummy variables" that have scope confined to the inside of their operators (sum/integral).  So, you can think of the $d$ as a prefix for the variable of integration.  
The choice of $d$ does suggest it is a limit of the $\Delta x_k$ in a Riemann sum.
