In this section we present the fundamental theorem of calculus. First, we must make a definition. A function F(x) is called an antiderivative of a function f (x) if f (x) is the derivative of F(x) ; that is, if F'(x) = f (x) . The antiderivative of a function f (x) is not unique, since adding a constant to a function does not change the value of its derivative:
(f (x) + c) = (f (x)) + (c) = |
It is true, however, that any two antiderivatives of a function f (x) will differ by a constant.
Why are antiderivatives a useful concept? Because they provide a shortcut for calculating definite integrals, as shown by the first part of the fundamental theorem of calculus.
Let f (x) be a function that is integrable on the interval [a, b] and let F(x) be an antiderivative of f (x) (that is, F'(x) = f (x) ). Then
f (x)dx = F(b) - F(a) |
Since the expression F(b) - F(a) is one we will encounter often, we will sometimes employ a special shorthand to simplify our equations:
F(x)|_{a} ^{b} = F(b) - F(a) |
Note that any antiderivative F(x) will give the same value for F(b) - F(a) , since any two antiderivatives differ only by a constant, which will cancel upon subtraction. The formula in the theorem may also be written
F'(x)dx = F(b) - F(a) |
From this, we see that taking the derivative of a function followed by a definite integral essentially gives us back the original function (or more precisely, the difference of its values at the limits of integration).
The second part of the fundamental theorem of calculus shows us that, in some sense, taking a definite integral followed by a derivative also gives back the original function. This is stated formally below.
Let f(x) be a function that is integrable on the interval [a, b] . Then for x in [a, b] :
f (t)dt = f (x) |
We therefore see that the derivative and integral, the two main tools of calculus, are essentially inverse operations on functions--they undo each other. This will prove to be incredibly useful for, among other things, calculating definite integrals without computing any clumsy Riemann sums.
Before putting the theorem to work, we briefly discuss why it is true. The first part of the theorem roughly states that in order to find the total change in the function F(x) from a to b , we must take a kind of sum (an integral) of the instantaneous rates of change (the derivatives) between a and b . The second part of the theorem says the instantaneous rate at which area is being added to the region under the graph as the right boundary of the region is extended is equal to the value of the function at the right boundary. Upon reflection, each of these facts makes sense intuitively.