Why can't we write integrals in a way analogous to sums, without adding a differential form : $\int_{x=a}^b f(x)$ instead of $\int_a^b f(x)dx$? It is my understanding that there is no mathematical significance to the "differential element" $dx$ at the end of an integral. Why hasn't the same convention as for sums with the sigma symbol been adopted ($\int_{x=a}^b f(x)$ instead of $\int_a^b f(x)dx$)? It seems clearer and more logical.
Edit: I understand that when using Lebesgue integrals the measure has to be indicated, but could this simpler notation at least work for integrals over sub-sets of $\mathbb{R}^n$ ?
Edit: I know that integrals are not the same as sums, but I think it would make sense to use similar notations. Also, there are striking similarities between properties of sums/series and properties of integrals: for instance, asymptotic comparisons, or convolution for functions vs Cauchy products for series.
 A: Notation is a matter of historical tradition and a matter of convenience, which is always dependent on context. This is true not only for the notation with integrals, but with the notation for sums as well, actually. For example, in elementary mathematics, we may denote $f(0) + f(1) + f(2)$ as $\sum_{n=0}^2f(n)$, but in higher mathematics, we are more inclined to using the notation $\sum_{n<3}f(n)$, which is more preferrable when working with ordinals in set-theory, or when doing discrete calculus. One may also write this as $\int{f\,\mathrm{d}c(x)}$ in the context of measure theory or time-scale calculus, where $c$ denotes the counting measure. The point is, depending on the application, we may want to modify a notation for the sake of convenience, but often, we also use a notation due to historical tradition.
In the context of real analysis, you are right: there genuinely is no significance to the symbol $\mathrm{d}x$. You can very much denote the Riemann integral of $f$ as $\int{f}$ if you want to. This is perfectly fine, and some authors actually do this. However, while this notation would be more convenient at the elementary level, it is desirable to include the symbol $\mathrm{d}x$ in higher mathematics for many reasons:

*

*In functional analysis, we would like to generalize the Riemann integral to the Riemann-Stieltjes integral with respect to an integrator $g$, which we denote as $\int{f\,\mathrm{d}g(x)}$. Here, the symbol definitely matters, because it denotes the integrator. Without it, one has no clue as to what the integrator actually is.

*Generalizing in a different direction, one may wish to generalize the Riemann integral to the Lebesgue integral, or even to the gauge integral, which has applications in physics. In this case, the symbol definitely matters, again, because it denotes which measure you are integrating with respect with. This actually is related to the Riemann-Stieltjes integral, and the full generalization is given by the Lebesgue-Stieltjes integral.

*Yet another direction of generalization is when generalizing Riemann integration to arbitrary manifolds. In there, we use the idea of differential forms, and the symbol there is absolutely necessary. Just for reference, look at the generalized Stokes' theorem, which states $$\int_M\,\mathrm{d}\omega=\int_{\partial{M}}\omega.$$ It is not possible to express the theorem conveniently in any other way if we do away with the differential symbol.

So while it is true that the symbol is completely unnecessary when simply doing Riemann integration, we choose to keep it, because it lends itself more easily to transitioning to the notation for generalizations, which do absolutely need the symbol, and we do this, even if it does make it a little bit harder for beginners. Oh, and by the way, you are not even obligated to use the integral symbol to denote the integral of a function. For example, in fractional calculus, it is not uncommon for a differintegral operator to be denoted as $J$. Then the differintegral of order $s$ of $f$ is simply denoted $J^s[f]$. You would never think this refers to an integral based on the notation, but there is nothing stopping you from adopting this convention, as long as you make it clear in your writings. The Daniel integral uses a similar type of notation. Again: it all boils down to context.
