Why do people interchange between $\int$ and $\sum$ so easily? One of the things I found curious in many texts is how in certain cases interchange the $\sum$ operator with $\int$. What are the "terms" for such a swap? I understand that integration in the early days was seen as an approximation of the area under the curve by using the very definition of multiplication and area to lend a hand with very small increments where the number of samples goes to infinity.
Beyond the original question, is this also the reason why we keep the right hand $dx$ (or any other infinitesimal variable), just to remind us of the origin because it "multiplies against the function", hence giving area. Or is there more to it?
Hints, answers, references to books... I'd appreciate anything you can give me.

IMHO, the integral which is the closest as a notion to the sum is Lebesgue integral. First, a sum is a Lebesgue integral with respect to an appropriate measure, i.e. 
$$
  \sum\limits_{i=1}^n a_i = \int\limits_1^n a(x)\;\mu(\mathrm dx)
$$
where $a(x)$ is any function with the only restriction $a(i) = a_i$ for $i=1,\dots,n$ and
measure $\mu$ is concetrated at points $1,2,\dots,n$ such that $\mu(1) = \mu(2) = \dots = \mu(n)$.
Since the sum is an object with many nice properties, it is always useful when the integral also shows similar properties. E.g. if $a_i\geq 0$ and 
$$
  \sum\limits_i a_i = 0
$$ 
then $a_i = 0$ for all $i$. For Lebesgue integral you have almost the same, namely if the function $f$ is such that $f(x)\geq 0$ and 
$$
  \int\limits_X f(x)\mu(\mathrm dx) = 0
$$
then the set $\{x:f(x)\neq 0\}$ is of zero measure $\mu$.
