5.3 Fundamental Theorem of Calculus
In this section we’ll see a most surprising connection between the antiderivative (section 5.1) and the definite integral (section 5.2). The name of this section is for a theorem that’s the key to solving definite integrals without explicitly resorting to limits. The Fundamental Theorem of Calculus makes computing definite integrals an exercise in finding antiderivatives. Via this theorem, many otherwise impossibly difficult calculations can be accomplished with ease.
Fundamental Theorem of Calculus
Suppose that f is bounded on the interval [a,b], and that F is an antiderivative of f, i.e., F’ = f. Then
Click here for a sketch of a proof of the Fundamental Theorem of Calculus.
Although F can be any antiderivative of f (the antiderivatives of f differ only in that they may have different constants of integration), for simplicity we always use the antiderivative with constant of integration equal to 0.
How remarkable that to calculate a definite integral, we need only evaluate the integrand’s antiderivative at the two endpoints of the interval!
We need to find the antiderivative of x3:
Notice we took the constant of integration, C, to be 0.
Applying the Fundamental Theorem of Calculus:
That’s the area under the graph of y = x3 (and above the x-axis) between x=0 and x=1.
Let’s look at example 1 again, but using some useful new notation which streamlines this process:
In words, “The integral of x3 from 0 to 1, equals x4/4 evaluated from 0 to 1, ... equals 1/4.”
To appreciate this technique, you only need attempt calculating this definite integral using the limit method of the last section. Or try redoing the exercises of the last section using this new technique – you’ll find they are now very easy calculations!
Now we can compute definite integrals whenever we know the antiderivatives of the terms of the integrand. Polynomials in particular are a breeze. Notice how we use the linearity property of the definite integral in the following integral calculations.
This is the area of one hump of the sine wave:
The result of this calculation is somewhat surprising!
Another form of the Fundamental Theorem of Calculus is easy to derive from the form we’ve just examined.
If F is defined by the integral:
Then F’(x) = f(x).
F is an antiderivative of f, and in fact it’s the one with the property that F(a) = 0, since
To see why, assume that G is an antiderivative of f. Then the Fundamental Theorem of Calculus says that
Since F and G only differ by a constant, G(a), they are both antiderivatives of f.
Many important functions can be defined in terms of definite integrals. For example, in Calculus II you’ll see how the natural log function can be defined by the integral:
But let’s not open that bag of worms until next semester!
This later form of the Fundamental Theorem of Calculus says (in other words) that
A generalization of this form of the Fundamental theorem is given by the following theorem. The proof of this theorem is an application of the first form of the Fundamental Theorem that we studied, and you should make sure that you understand the reasoning:
The Leibniz Integral Rule
Suppose that F is an antiderivative of f, so F’=f. Then by the Fundamental Theorem of Calculus,
Differentiating both sides of the equal sign (notice the chain rule being used on the right), we get:
Using the Leibniz Integral Rule
To see more examples of the Fundamental Theorem of Calculus, click here.
5.3 Exercises Do all of them! :-)