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

Fundamental Theorem of Integral Calculus (FTIC)

The relation in exercise 1 above is an example of the Fundamental Theorem of Integral Calculus, but for polynomials. We will now begin to show that this theorem also holds for any continuous

First Fundamental Theorem of Integral Calculus: Let be continuous on the closed interval and let Then

1. 
   In particular,
   
2. 
   In particular,
   
3. 
   is continuous on
   
4. 
   |
   

Proof:

1. 
   Let and hold fixed. Since is continuous on , it is right-hand continuous at Therefore, given
   

Choose Then,




In summary, Thus, In particular,

2. 
   Let and hold fixed. Since is continuous on , it is left-hand continuous at Therefore, given
   

Choose Then,




In summary, Thus, In particular,

3. 
   Since differentiability implies continuity, items a) and b) imply that is continuous on
   

4. 
   Also, from items a) and b), we see that Thus, 