Symbolically illustrate the definition of Riemann Sums.


  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

The algorithm is

  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, ..., an

  4. In each of the subintervals [ai-1, ai], we pick a number xi:
    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

Additional materials on Riemann Sums: