Symbolically illustrate the definition of Riemann Sums.
- a function y = f(x)
- an interval [a,b]
- a positive integer n
we want to find Riemann Sums corresponding to
- left-hand endpoints
- right-hand endpoints
The algorithm is
- Subdivide [a,b] into n subintervals of equal length.
- The length of each of these subintervals is
- We will label the endpoints of the subintervals:
a0, a1, a2, ..., an
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.
- We then form the Riemann Sum,
- 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:
Additional materials on Riemann Sums: