The Maple commands that we want to use are contained in the Student package. To access these commands, begin your session by entering
with(student);In addition to loading the commands in the package, this will list all the commands in the package. The commands that we will be using are
rightbox rightsum leftbox leftsum
middlebox middlesum value limit
The ``box'' commands produce a plot of the graph of a function showing
rectangles that fill the area under the graph. In combination with
value, a ``sum'' command computes the sum of the areas of
the rectangles plotted with the corresponding ``box'' command.
Let's illustrate these for the function 
between x
= 0 and 
. First, try the command
leftbox(1 + sin(x), x=0..Pi, 5);This plots the graph of f and 5 rectangles. Notice that the rectangles all have the same width,
. The height of each
rectangle is the height of the graph at the left endpoint of the
base of the rectangle. That's why this command has the name
leftbox. Now try
rightbox(1 + sin(x), x=0..Pi, 5);and
middlebox(1 + sin(x), x=0..Pi, 5);
We can see from any of these three plots, that using 5 boxes leaves a noticeable amount of ``white'' space underneath the curve and also creates ``green'' space above the curve. Let's try these commands again with more than 5 boxes (change 5 to larger numbers in these commands).
Of course, ``eyeballing'' it doesn't prove very much. What we need
to do is calculate the sum of the areas of these boxes. For this we
use value and the sum commands. The next commands
compute the sum of the areas of the 5 boxes that you plotted above
with the leftbox command. Try each of them.
value(leftsum(1 + sin(x),x=0..Pi,5));
evalf(value(leftsum(1 + sin(x),x=0..Pi,5)));
The second command gives a decimal value for the result of the first
command. Now try the evalf and value combination on each
of the box commands that you tried above. Caution: It is easy to make mistakes with parentheses, be sure each Maple command has the same number of left parentheses as right parentheses.