Introduction
This applet
allows you to experiment with Riemann sums and approximating the area
between the xaxis, the graph of y = f(x), the vertical
lines x = a and x = b. The user gets
to specify the function f, the values of a
and b and the number n of subintervals
[a,b] is to be divided into. The user also specifies
what kind of Riemann sum is to be used. The applet will then draw
the graph of the function, and the rectangles corresponding to the
partition and the kind of Riemann sum being used. For a more detailed
explanation of the options, read further.
Features
There are
seven areas on the applet the user can interact with. They are listed
below, along with an explanation. As the user puts the cursor into
a field, a brief explanation of what to do will be displayed in the
Messages area.
Viewing
Window Parameters.
In this
area, the user specifies the range for the x and yvalues. The default
is that x and y are between 10 and 10. Any number as well as any
expression resulting in a number can be used here. For example, when
trigonometric functions are studied, the user may prefer to use multiples
of pi such as 2*pi or 4*pi.
If "use yrange"
is checked (the default), then both the x and yrange values are used.
If it is unchecked, the applet will only use the xrange values and
compute the corresponding range for the yvalues.
The RESET
ZOOM button is used to reset the x and y ranges to their default
values.
Function
information
In this
area, the user specifies the function to use. The syntax is the same
as the one used in PlotSolve or the Java Math Pad. All the functions
built into the system can be used. Also combinations of these functions
by addition, subtraction, multiplication, division and composition
can be used. Finally, the derivatives of functions can be used. Here
are some possible examples:
 x^25*x
 xin(x^2)+cos(x)
 exp(x)ln(x)
 sin(exp(x^2))
 diff(sin(x^2))
Interval
[a,b]
The user
gets to specify the value of a, the left end point of
the interval of study;b, the right end point of the
interval of study and n, the number of subintervals.
Two methods can be used to specify a. Either click on
the corresponding field, and enter the value of a using
the keyboard. Alternatively, click on the corresponding field, then
click on the xaxis where a should be. The same possibility
exists for b, except that you click on the field corresponding
to b. n can only be entered using the
keyboard.
Riemann
Sum
In this
area, the user specifies what kind of Riemann sum to use from the
choice list. There are five choices:
 Upper. This
computes the upper Riemann sum. Under this option, the height of
the rectangles in each subinterval is the maximum value of the function
in each subinterval. This gives an upper bound for the area below
the graph.
 Lower. This
computes the lower Riemann sum. Under this option, the height of
the rectangles in each subinterval is the minimum value of the function
in each subinterval. This gives a lower bound for the area below
the graph.
 Midpoint. Under
this option, the height of the rectangles in each subinterval is
the value of the function at the midpoint of each subinterval.
 Left. Under
this option, the height of the rectangles in each subinterval is
the value of the function at the left point of each subinterval.
 Right. Under
this option, the height of the rectangles in each subinterval is
the value of the function at the right point of each subinterval.
Once the user
has selected the kind of Riemann sum to use, its value will be displayed.
Control
Buttons
The PLOT
button is used to plot the function after it has been entered.
The CLEAR ALL
button erases all the areas. The applet will look as if you had just
started it.
Plot
Area
This areas
displays the graph of the function, as well as the rectangles approximating
the area below the graph, according to the kind of Riemann sum being
chosen. The only interaction with this area is via the mouse. The
user can click near the xaxis to select a and b
as explained above.
Message
Area
There is
no interaction with this area. It is used to display hints on what
the user should do next as well as error messages if the wrong operation
is performed. Always look at what is displayed in this area.
User
Actions
The user can do the following:

Enter any function.

Specify the end points
a and b of the interval of study by
either entering their value using the keyboard or by clicking
on the xaxis.

Specify the number
of subintervals to use..

Specify the kind of
Riemann sum to use..
How
to Use the Applet
Use the applet as follows:

If necessary, change
the values in the Viewing Window Parameters area, though the default
values should be fine in many cases.

Enter a function.
Nothing can happen unless a function has been entered. Use the
same syntax as in the PlotSolve or the Java Math Pad applets.

Next, specify the
values of a, b and n
if the default values are not adequate.

Finally, specify the
kind of Riemann sum to use.
This applet was developed
by Dr. Philippe B. Laval,
at Kennesaw
State University.
This work was funded in part by:
The National Science Foundation # DUE9952568
The US Department of Education FIPSE #P116B00178