Second Fundamental Theorem of Calculus

1. The Mean Value Theorem For Integrals

   Let f be continuous on [a,b] then there is a c in [a,b] such that
   
   

    

   We define the average value of f(x) between a and b as
   
   

   Average Value =

    

   Example: 
   
   The average value of 
   
           sin(x) 
   
   between 0 and p is
   

          
   
   
   

2. The Second Fundamental Theorem of Calculus

   Let f be continuous on [a,b] then

   

   Proof:

   If then

   

   The average value theorem states that this integral is equal to

          

   for some value between x and x + h. Taking the limit produced

           f(c)
   
   

   Example:
   

          
   

   Example:
   

          
   
    Let
   
            
   
   and 
   
           u = x² 
   
   then