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

Proof:

3. 
   Constant Rule.
   

4. 
   Non-Negative Rule.
   

Proof: By the Max-Min Rule, 

5. 
   Positive Rule.
   

Proof: By the Max-Min Rule,

6. 
   Sum Rule.
   

Proof: Let Given for all partitions we have that

1. 
   
   
2. 
   
   
3. 
   
   

Let be a partition of such that and choose , the same for and

Note that . So, by a),b) and c), we obtain

Thus, Since was arbitrary, we conclude or 

7. 
   Scalar Multiple Rule.
   

Proof: Let and Given for all partitions of we have

1. 
   .
   
2. 
   .
   

Let be a partition of such that Note that . So, by a) and b), we obtain

Thus, Since was arbitrary, we conclude that or . 

8. 
   Linear Rule. This is equivalent to the Sum Rule together with the
   

Scalar Multiple Rule.

9. 
   Non-Decreasing Rule.