Definition (Definite Integral): Let be continuous on the closed interval

Riemann-Darboux Condition Theorem: Let be bounded on The following are equivalent.

1. 
   is Riemann/Darboux integrable over
   
2. 
   
   
3. 
   
   
4. 
   If and where then regardless of the chosen
   
5. 
   If and where then regardless of the chosen
   

Example: Not all functions are Riemann integrable. Consider called the Dirichlet function. Let be a partition of the interval Now,

1. 
   Since in any interval there is an irrational number, we have Therefore,
   
2. 
   Since in any interval there is a rational number, we have Therefore,
   

Thus, for the Dirichlet function, we have shown that and for any partition of This says that is not Riemann integrable on

The following theorem shows us that not all functions have to be continuous in order to be integrable.

Theorem: Every is integrable on . Let be any partition of . Then, and We then have

In summary, we have which implies is integrable over 

Example: Using the above definition of the Riemann integral, show that the area under the curve from to where  and , is . That is, show
Proof: First let us consider some preliminary remarks so that the proof is easier to understand. Recall, for ,

If then

We now begin our proof.
Let be any partition of . In what follows think of as and as where appropriate. Since , then, for we have

But . So,

(*) .
Observe that,
(**) .

This series is called a telescoping series because of the way the terms in the sum cancel.

Summing (*) from to we obtain
(***)
Using our observation in (**) and dividing by (***), we finally obtain from (***)

Since

Thus,

This clearly implies that




Exercises: Let ℝ. Use the procedure in the above example to determine 1)-3).

1. 
   if and is even.
   
2. 
   if and is odd.
   
3. 
   if