I am trying to properly understand the first fundamental theorem of calculus but there are several little details that are giving me a lot of trouble. So I figured here I would describe my understanding of it and explain what gives me trouble.
Suppose I let the function $A(x)=\int_a^xf(t)dt$ define the area bound by a curve $f(t)$ and the lines $t=a$ and $t=x$, such that $a\leq x$, and the horizontal axis. $A(x)$ is a function of $x$. The fundamental theorem of calculus states, that $\frac{d}{dx}A(x)=\frac{d}{dx}\int_a^xf(t)dt=f(x)$.
We say that $A(x)$ is the antiderivative of f(x), does this mean that it is the indefinite integral of f(x)? If so, is the theorem using a definite integral of f(t) to get to un indefinite integral of $f(x)$?
Is $f(t)$ the same function as $f(x)$? What I mean by this, is that if $f(t)=t\times cos(2t^2-3t)$, would $f(x)=x\times cos(2x^2-3x)$? If so, is this an assumption we need to make to define the fundamental theorem of calculus, or is it a consequence of it?
Does the fact that both functions are given the letter $f$ meant that they are both the same? If this is the case, could we maybe not use $t$ is a variable, and use $x$ instead? I know it might get a little confusing since $x$ is also one of the boundaries of the integral, but could it be done in principle?
This is I think the most crucial question I have. The first part of the fundamental theorem of calculus is used to prove that integration and differentiation are inverses of each other. But my problem is that the area that we are finding is bound by the function $f(t)$, not the function $f(x)$. Essentially, what I think the theorem is saying, is that the derivative of the function that gives us the area under the curve $f(t)$ on an interval between $t=a$ and $t=x$ is $f(x)$. If we really wanted to prove that integration is the inverse of differentiation, wouldn't we have to prove that the area function gives us the area under $f(x)$?
I guess that the crux of my issue is that I struggle to figure out how the two variables $t$ and $x$ relate to each other. Because if both $f(t)$ and $f(x)$ can be used to find area $A(x)$, then they must be related to each other in some way right?