Indeed you can see $\int_a^bf(x)dx$ (1) as an infinite sum of rectangles where $f(x)dx$ is a rectangle of width $dx$ and height $f(x)$. This corresponds to the Riemann integral .
However, this is actually one interpretation (rectangles) of one example (the formula (1)), using one definition of an integral (Riemann integral). You can give other forms, and you can look for other integrals (Lebegues, Itô, etc) and work inside other theories, as well as you can create your own definition and your own theory.
Examples and images are important to get a feeling of what a mathematical object can be, however, as you go further in mathematics, you'll enrich your inner feeling of it as you add new interpretations and examples. Can can see an example of a projection; and what matters to the mathematician is not the projection but the whole set.
Another few things matter too. You wrote the formula (1) but without giving what are $a,b,x$ or the theory in which you're working in. Most of the time it's not given, and many mathematicians don't be specific about which theory they are working in. Most of the time it'll be in ZFC + first order logic.
I would like to share few other point of views. The form you wrote (1) (where $a,b$ are constants and $x$ a variable) is also a value. It can have a dimension or not.
For $a,b\in \mathbb X$ and $x \to f(x): \mathbb X\to \mathbb X$ and if $\mathbb X$ is $\mathbb R$ then we can see $a,b,x,f(x)$ as lengths and (1) as a surface.
You can have $\mathbb X$ as a set of vectors, and $a,b,f(x),x$ as vectors; eg., each vector represents as set of particles (eg., a gaz). Then $\mathbb Z$ will be $\mathbb R^n \forall n\in \mathbb N$ .
$\int _a^zf(x)dx$ is a function of $z$, and this is definitely different than a value. It can also be a function of $a,b,z$ where all three are variables.
(1) could be an Itô integral , and will be interpreted as a random variable, or the path of a stochastic process in an unspecified dimension.
(1) could be a set of proofs of a sentence $\mathbb X$ [3a, 3b]
I would like to discuss a point that is extremely commonly encountered. There is often a confusion between function: $f$ or $f(\cdot)$ and its value at $x$: $f(x)$. They are really not the same. However, usually if $x$ is a variable, we'll see $f(x)$ as a function and if $x$ is a constant, $f(x)$ as a value, but this is only a convenience. A good way to see that is that $f(x)$ is the result of a projection.
You'll see the same duality with integrals: if can be a function or a value. We write the function, but it's actually "how we get the value". An integral over a domain is also a projection, only more explicit than that of $f(x)$. Each time we do a sum or a projection, we loose data, and the result is having as many dimension less as integral(s) (we usually write one $\int$ instead of many $\int \int \int \ldots$). You'll see that a lot in quantum physics; where an integral will be a measure, and as any measure, it's a projection. You'll also loose data as to get the measure, as well as you loose "how to get the value" (the integral as function), when you compute over a domain and get the value (the integral as value).
Maybe the point is that I encourage anyone to be careful and critical about the taught mathematics, since they are often simplified, interpreted; and I really encourage to a personnal interpretation.
Another example, is the developped and factorized forms of a polynom. We put the sign equal between them, but it only means that their value is the same, but they are not equal. One form has more information (the factorized one). And the whole process of transformation is also information.