I am learning about the fundamental theorem of calculus at the moment. It makes perfect sense to me that $$\int_a^b f(t)\;dt=F(b)-F(a)\tag1$$
What makes me confused is this different variables x and dt in the following:
$$F(x)=\int_a^x f(t)\;dt\tag2$$
So I tried to work through it and I got some idea, but I am not sure if this is correct:
I. Both (1) and (2) are actually about a function f(t)
II. In (1) we are trying to evaluate the area under f(t) over an intervall [a,b]
III. (This is where I am not sure and get confused) In (2) we are evaluating the area under f(t) over an intervall [a,x]
IV. (Even more insecured) that x of the statement III before has actually nothing to do with the often used "f(x) function". In statement (2) F(x) does not mean "a function F of x" but it is a value x for the function F(t).
Is this correct so far? So the main question is about the last one: does F(x) mean in this case "the value of x for t in F(t)" ?
Edit: To make my question more clear: What is F(x) ? I mean indefinite integrals states that when I form the antiderivative of f(t) I will receive F(t); If I then want to know the specific area under a curve between to points I fill in those values in my function F(t). How comes that [in (2)] suddenly we receive a function with a new variable? Where does the t disappears to ?
I have tried to explain my problem in three pictures, the function used is the simple one f(t)=t, so F(t)=1:
I. They are both about a function $f$, yes. The 'name' of the variable doesn't really matter, it could be $f(x)$, $f(t)$, whatever, as long as the variable in $f(.)$ is a real-valued variable.
II. Yes. You are integrating the function $f(t)$ over the interval, $t \in (a,b)$.
III. In (2) you are defining a new function, $F(x)$. The definition for this function states that, at any given real number $x$, the value of the function $F(x)$ is equal to that interval... in other words, $F(x)$ is defined as the value obtained from integrating the function $f(t)$ over the interval $t \in (a,x)$.
IV. No. $F(x)$ is certainly a function of $x$, and $f(t)$ is certainly a function of $t$. In (2), the 'naming' of $x$ and $t$ are completely arbitrary.
You are first given the function, $f$, and are (probably) also told it's defined on some interval (at least) containing $(a,b)$. So, given any real number - whether it's $x$, $t$, $a$, $b$, $q$?, anything defined in that interval - you can compute the value of $f$ at that point, $f(x)$, $f(t)$, etc.
(1) states that the integral of $f$, when taken on the interval $(a,b)$ - the area under the curve $f(t)$ on the interval $t \in (a,b)$ - is equal to the difference between two numbers. Specifically, these two numbers are the values of some function $F$ at those points defining the interval, $a$ and $b$. From this definition alone, you do not know what $F$ is, just that some function $F$ exists and that the integral is equal to $F(b) - F(a)$.
(2) provides a definition for that function, so for any given real number $x$, you know how to find the value $F(x)$. So since $a$ and $b$ are real numbers, you can use $x=a$ and $x=b$ to find $F(a)$ and $F(b)$. The reason $x$ and $t$ are separate variables in (2) is that you're defining what the function's value $F(x)$ is for a specific $x$, but this value is computed by integrating the function $f$ over an interval of $(a,x)$, so $t$ is used as the variable of integration.