Can a function describe an area on a graph like an integral? Am I correct to say:
A difference between an integral and a function is: that an integral can describe an area on a graph, while a function can't?
Or am I completely off course here?
 A: A function $f : A \rightarrow B$ is just an object which, given any $a \in A$ then $f(a) \in B$.
The graph of a function $\{(a,f(a))|a\in A\}$ in the case of $f : \mathbb R \rightarrow \mathbb R$ can be viewed as a curve but that's just an illustration.
An anti-derivative, say $F(t) = \int_0^{t} f(x) \mathrm{d}x$ is just another function $F : \mathbb R \rightarrow \mathbb R$.
While it is true that say, $F(3)$ gives the area under the curve $f$ defines between $0$ and $3$ on the x-axis.. but this idea of 'area' is just an intuitive guide to help motivate the definitions of integral and similar.
The integral $\int_{a}^{b}$ could be thought of as a map that takes $(\mathbb R \rightarrow \mathbb R) \rightarrow \mathbb R$ but that is not usual and it can also just been seen as a notation which only has meaning when completed (given an integrand).
A: You can view the integral as a function too, assuming it is defined. For instance, if you fix an interval $I=[a,b]$, and consider all functions that are integrable on $I$, you can view the integral as a function that maps each of these functions to a real number, that represents the 'area under the curve'.
Or, if you fix a function $f$ on the same interval, you can view the integral from $a$ to $x$ (where $a\leq x\leq b$) as a new function of $x$, defined on the interval, and the value at each $x$ would represent the area under the curve up to $x$.
