Although I have worked with integrals for years, suddenly a very elementary question about the intuition behind it appeared to me. It is basically about constructing (indefinite) integrals in two steps.
Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is a function. The indefinite integral is given as $\int f(x) \text{d}x$. Obtaining the indefinite integral from a function $f$ can be regarded as an operation that maps a function to another function, namely its anti-derivative. Based on the Riemannian interpretation, this has always worked for me.
Now, let's divide this procedure into two steps: First, we obtain the differential of the function $f$, which is given as $dF(x,\text{d}x) := f(x)\text{d}x$. We can view this differential as a function on two variables that is linear in $\text{d}x$. After that, we integrate this differential to obtain $\int f(x) \text{d}x$. My question is: what is the intuition behind both steps? The first step I consider as obtaining a linear approximation of $f$ at any point $x$. But, given that interpretation, what does taking the integral actually mean when the $\text{d}x$ is viewed as a variable of the differential?