We set it up initially as f(t) because we're using x for a different purpose (area under the curve) and it would create a logical contradiction for x to be both the independent variable that produces the curve and the independent variable that produces the area under the curve.
It can operate on t or x or any other variable or constant within its domain. The key to understanding this is to realize that f, by itself, is a function. Some people have a hard time with this duality of f(t) and f(x). We're trying to show that if we have a function F(x) that provides the area under the curve f(t), the derivative of F(x) is f(x). If you get that much firmly in mind, the rest should be easier, but there are a couple of other points of confusion. We're given a function f(t) and asked to think about another function, not displayed, that is the area under the curve, not the value displayed on the curve. The graph doesn't show F(x) at all in fact, it doesn't have an x-axis. As x moves to the right, this area increases, even if f(t) is decreasing. Instead, F(x) is the area under this graph between point a and point x. So the first thing I would offer in trying to understand this better is to get a clear picture that this graph does not depict F(x). In this case, we're studying a function F(x) but looking at a graph of a different function, f(t).
Part of the problem is that in almost all our other work we're looking at a graph of the function we're studying.
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