Graphically illustrate = the=20 definition of Riemann Sums for the function,

The blue line in the graph is part of the * x-axis*.=20

**Note.** The definition of **Riemann Sums** will be given in =
the=20
development that follows.=20

- First, we pick some positive integer
**n**. For our = illustration, we=20 shall pick**n**=3D 10. - We now subdivide the interval interval into
**n**equal = subintervals.=20 - Each of the new subintervals has length=20
- We will label the endpoints of the new subintervals:=20
**a**_{0}, a_{1}, a_{2}, ...,=20 a_{10}which is called a

**partition of = [a,b]**.=20 - In each of the subintervals =
**[a**,=20 we pick a number_{i-1}, a_{i}]**x**and draw a line segment = perpendicular to=20 the_{i}from the point**x**-axis**(x**to a = point on=20 the graph of the function, =_{i},0)**(x**.=20_{i}, f(x_{i})) - As in this animation, we then construct rectangles which have the =
line=20
segments as their height and the subintervals as their base.=20
If each

**f(x**then the area of = the=20_{i}) > 0**i**rectangle is=20^{th}and the sum of the areas of the rectangles is then:=20

More generally, we do not require that=20

**f(x**as we define=20_{i}) > 0A **Riemann Sum of = f over=20 [a, b]**is the sum=20

If you want to view some additional graphs illustrating Riemann Sums =
with=20
different values of **n** and different choices of =
**x _{i}**'s,=20
then make your choices from the following two groups of options:=20