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The Second Fundamental Theorem of Calculus

Consider the function

g(x) = f (t)dt    

for the graph of f drawn below:

Figure %: Graph of f

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.

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