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

Remark: The limit satisfies all the usual properties of limits.
Remark: For , it is convenient sometimes to use one of the following:

1. 
   . The left end points of the subintervals corresponding to the regular partition
   
2. 
   The right end points of the subintervals corresponding the regular partition
   
3. 
   The midpoints of each subinterval ,
   
4. 
   is a maximum in assuming f is continuous on .
   

The corresponding Riemann sum is then denoted by called the upper Darboux sum of f over ³.

5. 
   is a minimum in assuming f is continuous on .
   

The corresponding Riemann sum is then denoted by , called the lower Darboux sum of f over ⁴
Note that we always have
There is an alternative but logically equivalent definition of the integral that is conceptually easier to understand and that uses Darboux sums.