The applet below adds up the areas of a set of rectangles to approximate
the area under the graph of a function. You have a choice of three different
functions. In each case, the area approximated is above the interval [0, 5]
on the xaxis.
You have a choice between using rectangles which
touch the curve at the top of the rectangle's left side or the top of the rectangle's
right side. The former choice gives what are called lefthand sums,
the latter choice gives righthand sums.
Experiment with changing the number of rectangles, changing the function itself,
and switching between lefthand sums and righthand sums.

For the parabolic function y = (x^{2} + 5)/6, use the applet to find the
best upper bound for the area under the curve that you can. Likewise, find the
best lower bound.

Repeat step 1 for the function y = sqrt(25  x^{2}). What property must
a function have to be sure that the lefthand sums give an upper bound for the
area under the curve and the righthand sums give a lower bound? What property
assures the opposite? Does the function y = 2*cos(x) have either necessary property?
What can you say about upper and lower bounds for the area in this case?

Define the function f(x) = (x^{2} + 5)/6 in your helper application. Then
calculate the lefthand sum with n = 5 rectangles. Note that the heights of the
five rectangles whose areas you want are f(0), f(1), f(2), f(3), and f(4). Your
calculation should give the same result as the applet. To do another check of the
applet's accuracy, use your helper application to calculate the lefthand sum using
n = 10 rectangles. This time the rectangle heights are f(0.0), f(0.5), f(1.0), ... , f(4.5)
and each base is of length 1/2. Compare your result to the applet's result.

Repeat step 3 for righthand sums. Compare your results to those of the applet.

Suppose we want to approximate the area under a function f(x) above the interval
[a, b] on the xaxis using a lefthand sum with n rectangles,
all with equal bases.
Tell why each base must have length delx = (b  a)/n.
Thus the base of the first rectangle extends from a to a + delx
along the xaxis.
The base of the second rectangle extends from a + delx to a + 2*delx
along the xaxis.
In terms of parameters a and delx, the i th rectangle's
base extends between what two x values? (Your answer will have an i in it.)
What is the formula for the area of the i th rectangle in the lefthand sum?
(Your answer will depend on a, delx, and i.)

In the helper application file that accompanies this module, you will find commands
based upon the results of step 5 above that
will calculate a lefthand sum for input values a, b, and n.
Check that these commands give lefthand sums that match those of our applet for each
number of rectangles n given in the applet. Use the function
f(x) = (x^{2} + 5)/6.

Use the lefthand sum commands to calculate tendecimalplace approximations to the area
under the curve
f(x) = 1/x above the interval [1,3] on the xaxis. Do approximations for n = 5 subintervals,
for n = 10 subintervals, for n = 20 subintervals, and for n = 40 subintervals. Save your
results in the table in your helper application "notebook." You will need these results in
Parts 4 and 5 of this module.

Modify the commands in your helper application file that compute lefthand sums
so they will instead calculate righthand sums for input values a,
b, and n.
Check that your new commands give righthand sums that match those of our applet for each
number of rectangles n given in the applet. Use the function
f(x) = (x^{2} + 5)/6.

Use the righthand sum commands to calculate tendecimalplace approximations to the
area under the curve
f(x) = 1/x above the interval [1,3] on the xaxis. Do approximations for n = 5 subintervals,
for n = 10 subintervals, for n = 20 subintervals, and for n = 40 subintervals. Save your
results in the table in your helper application "notebook" for use in
Parts 4 and 5 of this module.