Calculus/Fundamental Theorem of Calculus

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Fundamental Theorem of Calculus

The fundamental theorem of calculus is a critical portion of calculus because it links the concept of a derivative to that of an integral. As a result, we can use our knowledge of derivatives to find the area under the curve, which is often quicker and simpler than using the definition of the integral.

Contents

Mean Value Theorem for IntegrationEdit

We will need the following theorem in the discussion of the Fundamental Theorem of Calculus.

Mean Value Theorem for Integration

Suppose   is continuous on   . Then   for some   in   .

Proof of the Mean Value Theorem for IntegrationEdit

  satisfies the requirements of the Extreme Value Theorem, so it has a minimum   and a maximum   in   . Since
 
and since
  for all   in   ,
we have

 

 

 

 

 

Since   is continuous, by the Intermediate Value Theorem there is some   with   in   such that

 

Fundamental Theorem of CalculusEdit

Statement of the Fundamental TheoremEdit

Suppose that   is continuous on   . We can define a function   by

 

Fundamental Theorem of Calculus Part I Suppose   is continuous on   and   is defined by

 

Then   is differentiable on   and for all   ,

 

When we have such functions   and   where   for every   in some interval   we say that   is the antiderivative of   on  .

Fundamental Theorem of Calculus Part II Suppose that   is continuous on   and that   is any antiderivative of   . Then

 
 
Figure 1

Note: a minority of mathematicians refer to part one as two and part two as one. All mathematicians refer to what is stated here as part 2 as The Fundamental Theorem of Calculus.

ProofsEdit

Proof of Fundamental Theorem of Calculus Part IEdit

Suppose   is in   . Pick   so that   is also in   . Then

 

and

  .

Subtracting the two equations gives

 

Now

 

so rearranging this we have

 

According to the Mean Value Theorem for Integration, there exists a   such that

  .

Notice that   depends on   . Anyway what we have shown is that

  ,

and dividing both sides by   gives

  .

Take the limit as   we get the definition of the derivative of   at   so we have

  .

To find the other limit, we will use the squeeze theorem. The number   is in the interval   , so   . Also,   and   . Therefore, according to the squeeze theorem,

  .

As   is continuous we have

 

which completes the proof.

Proof of Fundamental Theorem of Calculus Part IIEdit

Define   . Then by the Fundamental Theorem of Calculus part I we know that   is differentiable on   and for all  

 

So   is an antiderivative of   . Since we were assuming that   was also an antiderivative for all  ,

 
 
 

Let   . The Mean Value Theorem applied to   on   with   says that

 

for some   in   . But since   for all   in   ,   must equal   for all   in   , i.e. g(x) is constant on   .

This implies there is a constant   such that for all   ,

  ,

and as   is continuous we see this holds when   and   as well. And putting   gives

  .

Notation for Evaluating Definite IntegralsEdit

The second part of the Fundamental Theorem of Calculus gives us a way to calculate definite integrals. Just find an antiderivative of the integrand, and subtract the value of the antiderivative at the lower bound from the value of the antiderivative at the upper bound. That is

 

where   . As a convenience, we use the notation

 

to represent   .

Integration of PolynomialsEdit

Using the power rule for differentiation we can find a formula for the integral of a power using the Fundamental Theorem of Calculus. Let   . We want to find an antiderivative for   . Since the differentiation rule for powers lowers the power by 1 we have that

 

As long as   we can divide by   to get

 

So the function   is an antiderivative of   . If   is not in   then   is continuous on   and, by applying the Fundamental Theorem of Calculus, we can calculate the integral of   to get the following rule.

Power Rule of Integration I

  as long as   and   is not in   .

Notice that we allow all values of   , even negative or fractional. If   then this works even if   includes   .

Power Rule of Integration II

  as long as   .

Examples

  • To find   we raise the power by 1 and have to divide by 4. So
  .
  • The power rule also works for negative powers. For instance
  .
  • We can also use the power rule for fractional powers. For instance
 
  • Using linearity the power rule can also be thought of as applying to constants. For example,
  .
  • Using the linearity rule we can now integrate any polynomial. For example
 

ExercisesEdit

1. Evaluate   . Compare your answer to the answer you got for exercise 1 in section 4.1.

 

2. Evaluate   . Compare your answer to the answer you got for exercise 2 in section 4.1.

 

3. Evaluate   . Compare your answer to the answer you got for exercise 4 in section 4.1.

 

Solutions

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Fundamental Theorem of Calculus