understanding integrals I heard that 

An integral 
1: Is a function
2: That can be represented as a curve , and
3: Also that can be understood as the area behind a curve and the x axis.
4: it is also possible to understand them as a sum of infinite areas infinitely small
Questions
(A) Are there more representations of the idea of an Integral?
Is that is so,
(B) Can we have a sort of categories that help understanding how to represent the idea of an integral
(C) Can we propose categories for a common understanding?
For example:
Geometric: Area behind a Curve
Analytic:  The integral plus a constant is equal to derivative.
(D) Does this means that there is something absolutely abstract that we approach by representations and those representations are categorized in areas of knowledge that also serves to helping us understanding our understanding of the way our minds approach the world?
 A: The basic idea behind the notion of "integral" is the following:
In ${\mathbb R}^n$, $\>n\geq1$, or some reasonable part of it, an  "intensity" $f$ is given which varies from point to point. This means we have a function $$f:\quad \Omega\to{\mathbb R},\qquad {\bf x}\mapsto f({\bf x})\ ,$$
where $\Omega\subset{\mathbb R}^n$ is the domain of $f$. In the simplest case $f$ is a function defined on some interval $[a,b]\subset{\mathbb R}$.
Given some subset $A\subset\Omega$ (an interval, a cube, a ball, or some more complicated geometrical object) we are interested in the "total effect" this variable intensity has when restricted to $A$. This "total effect" is denoted by
$$\int\nolimits_A f({\bf  x})\ {\rm d}({\bf x})$$
and is called the integral of $f$ over $A$.
This notion should have the following properties:
$$\int\nolimits_A \bigl(f({\bf  x})+g({\bf  x})\bigr)\ {\rm d}({\bf x})=\int\nolimits_A f({\bf  x})\ {\rm d}({\bf x})+\int\nolimits_A g({\bf  x})\ {\rm d}({\bf x})\ ,\quad \int\nolimits_A \lambda f({\bf  x})\ {\rm d}({\bf x})=\lambda\int\nolimits_A f({\bf  x})\ {\rm d}({\bf x})\ ,$$
$$\int\nolimits_{A\cup B} f({\bf  x})\ {\rm d}({\bf x})=\int\nolimits_A f({\bf  x})\ {\rm d}({\bf x})+\int\nolimits_B f({\bf  x})\ {\rm d}({\bf x})$$
when $A$ and $B$ are essentially disjoint, and finally
$$\int\nolimits_A 1\ {\rm d}({\bf x})={\rm vol}(A)\ .$$
A lot of work is needed to establish this notion for $f$'s and $A$'s as general as possible and then to create algorithms for computing the integral when $f$ and $A$ are given in some explicit way. In the one-dimensional case we have the fundamental theorem of calculus, which allows to compute the integral by means of "primitives" (antiderivatives) of the given function $f$.
A: I am trying a totally different route here. My suggestion is to read about the history of Newton and Leibniz regarding their discovery about calculus. Perhaps that might shed some light on your (broad) questions as how to answer them. Quite often when going into the history of a topic pertaining the question, some very fine answers may be obtained.
A: Germán Muñoz Ortega offers a very interesting point of view that can help proposing a conceptual framework.  The author purposes in his paper "Elementos de enlace entre lo conceptual y lo algorítmico en el Cálculo" in the Relime Magazine of july 2000 to make a study of the conceptual and algorithmic aspects of the integration processes.
Following this idea, one can purpose that exists at least this two major categories:
1:  Algorithmic approach
2:  Conceptual approach
And we can go further
1:  Algorithmic (we can approach the comprehension of merely the processes themselves as a way of understanding integrals by learning how to compute them)
1.1:  Computable processes (If is the case we can make a student recognize some integrals and compute their solutions, and this is a kind of understanding)
1.2:  Not computable processes (Even if the case is that the process is not computable, or saying in a different way, that the integral does not have a solution is a level of understanding)
2:  Conceptual approach (Different non algorithmic ideas related to integrals)
2.1:  Historical epistemic approaches before S XVIII (The kind of answers we can describe or arrive using integrals, for example:  variation, accumulations, summations, successive derivation,etc)
2.2:  Meaning of the integral object it self as calculus of numeric functions  (S XIX and further). 
