Problem:
Graphically illustrate the definition of Riemann Sums for the function, y = f(x) with domain [a, b], whose graph is

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.

Visualization:

  1. First, we pick some positive integer n. For our illustration, we shall pick n = 10.

  2. We now subdivide the interval interval into n equal subintervals.

  3. Each of the new subintervals has length

  4. We will label the endpoints of the new subintervals:

    a0, a1, a2, ..., a10

    which is called a partition of [a,b].

  5. In each of the subintervals [ai-1, ai], we pick a number xi and draw a line segment perpendicular to the x-axis from the point (xi,0) to a point on the graph of the function, (xi, f(xi)).

  6. 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(xi) > 0 then the area of the ith rectangle is

    and the sum of the areas of the rectangles is then:

    More generally, we do not require that f(xi) > 0 as we define

    A 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 xi's, then make your choices from the following two groups of options:

Choose number of subdivisions:

n = 10
n = 20
n = 40
Type:

xi is right-hand endpoint of subinterval
xi is left-hand endpoint of subinterval
xi is midpoint of subinterval
|f(xi)| is the maximum of |f| on the subinterval (circumscribed rectangles)
|f(xi)| is the minimum of |f| on the subinterval (inscribed rectangles)

Note that the Riemann sum when each xi is the right-hand endpoint of the subinterval [ai-1, ai] is

when each xi is the left-hand endpoint of the subinterval [ai-1, ai] is

and when each xi is the left-hand midpoint of the subinterval [ai-1, ai] is

.

Summary of the material above.