Consider the function
|g(x) = f (t)dt|
for the graph of f drawn below:
In words, what does the function g(x) represent on the graph above?
The function g(x) represents the area under the graph of f between a and x. Thus, g(a) = 0, since the area between a and a is zero. According to the second fundamental theorem of calculus,
|f (t)dt = f (x)|
This tells us two things. First, an area function of f is always an antiderivative of f. Second, differentiation and integration are inverse operations, because integrating f and then differentiating it yields f again.