The second FTOC (a result so nice they proved it twice?) introduces a totally bizarre new kind of function. A function defined as a definite integral where the variable is in the limits. Weird! The variable in the integrand is not the variable of the function.
I think many people get confused by overidentifying the antiderivative and the idea of area under the curve. Since that's the point of the FTOC, it makes it hard to understand it.
This sketch tries to back it up. Let's define one of these functions and see what it's like. In this sketch you can pick the function f(x) under which we're finding the area. You can pick the starting point, and then the sketch calculates the area under f from the starting point to the value x that you pick. That area is the value of F(x).
Play with the sketch a bit. What do you notice? How does the starting value affect F(x)? Can you predict F(x) before you trace it out?