Given:

1. a function y = f(x)
2. an interval [a,b]
3. a positive integer n

we want to find Riemann Sums corresponding to

1. left-hand endpoints
2. right-hand endpoints
3. midpoints

1. Subdivide [a,b] into n subintervals of equal length.
2. The length of each of these subintervals is

3. We will label the endpoints of the subintervals:
   a₀, a₁, a₂, ..., a_n where

4. In each of the subintervals [a_i-1, a_i], we pick a number x_i: depending upon whether we want to use left-hand endpoints, right-hand endpoints or midpoints, respectively.

5. We then form the Riemann Sum,

6. Substituting from 4. above, we get the Riemann Sums using

   • left-hand endpoints:
     

     which is also equal to

     which is very useful in computations

   • right-hand endpoints:
     

   • midpoints:
     

• Drill in finding Riemann Sums of linear functions (using IBM TechExplorer)
• Graphical illustration of Riemann Sums
• Using a calculator to evaluate Riemann Sums