Graphically illustrate the definition of Riemann Sums for the function,

The blue line in the graph is part of the * x-axis*.

**Note.** The definition of **Riemann Sums** will be given in the
development that follows.

- First, we pick some positive integer
**n**. For our illustration, we shall pick**n**= 10. - We now subdivide the interval interval into
**n**equal subintervals. - Each of the new subintervals has length
- We will label the endpoints of the new subintervals:
**a**_{0}, a_{1}, a_{2}, ..., a_{10}which is called a

**partition of [a,b]**. - In each of the subintervals
**[a**, we pick a number_{i-1}, a_{i}]**x**and draw a line segment perpendicular to the_{i}from the point**x**-axis**(x**to a point on the graph of the function,_{i},0)**(x**._{i}, f(x_{i})) - As in this animation, we then construct rectangles which have the line
segments as their height and the subintervals as their base.
If each

**f(x**then the area of the_{i}) > 0**i**rectangle is^{th}and the sum of the areas of the rectangles is then:

More generally, we do not require that

**f(x**as we define_{i}) > 0A **Riemann Sum of f over [a, b]**is the sum

If you want to view some additional graphs illustrating Riemann Sums with
different values of **n** and different choices of **x _{i}**'s,
then make your choices from the following two groups of options:

Note that the Riemann sum when each **x _{i}** is the right-hand
endpoint of the subinterval

when each **x _{i}** is the left-hand endpoint of the subinterval

and when each **x _{i}** is the left-hand midpoint of the
subinterval