5.3  Fundamental Theorem of Calculus

In this section weíll 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 thatís 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.,  Fí = 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 integrandís antiderivative at the two endpoints of the interval!

Example 1:    

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:

 Thatís the area under the graph of y = x3 (and above the x-axis) between x=0 and x=1.

Letís look at example 1 again, but using some useful new notation which streamlines this process: 

In words, ďThe integral of  x3  from 0 to 1, equals x4/4 evaluated from 0 to 1, ... equals 1/4.Ē

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 Ė youíll 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.

Example 2:   

Example 3:   

Example 4:   

This is the area of one hump of the sine wave:


The result of this calculation is somewhat surprising!

Example 5:   

Another form of the Fundamental Theorem of Calculus is easy to derive from the form weíve just examined.

If  F  is defined by the integral:

Then  (x) = f(x).  

 F is an antiderivative of f, and in fact itís 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 youíll see how the natural log function can be defined by the integral:

But letís 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.    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:

Example 6:    

Using the Leibniz Integral Rule

To see more examples of the Fundamental Theorem of Calculus, click here.

5.3 Exercises   Do all of them!   :-)

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Rafael Espericueta
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