Symbolically illustrate the definition of Riemann Sums.

Given:

- 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
- midpoints

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:
**a**_{0}, a_{1}, a_{2}, ..., a_{n} -
In each of the subintervals
**[a**, we pick a number_{i-1}, a_{i}]**x**:_{i}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:
- midpoints:

- left-hand endpoints:

Additional materials on Riemann Sums: