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 x-axis.

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 left-hand sums, the latter choice gives right-hand sums.

Experiment with changing the number of rectangles, changing the function itself, and switching between left-hand sums and right-hand sums.

1. For the parabolic function y = (x² + 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.
2. Repeat step 1 for the function y = sqrt(25 - x²). What property must a function have to be sure that the left-hand sums give an upper bound for the area under the curve and the right-hand 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?
3. Define the function f(x) = (x² + 5)/6 in your helper application. Then calculate the left-hand 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 left-hand 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.
4. Repeat step 3 for right-hand sums. Compare your results to those of the applet.
5. Suppose we want to approximate the area under a function f(x) above the interval [a, b] on the x-axis using a left-hand 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 x-axis. The base of the second rectangle extends from a + delx to a + 2*delx along the x-axis. 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 left-hand sum? (Your answer will depend on a, delx, and i.)

6. 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 left-hand sum for input values a, b, and n. Check that these commands give left-hand sums that match those of our applet for each number of rectangles n given in the applet. Use the function f(x) = (x² + 5)/6.
7. Use the left-hand sum commands to calculate ten-decimal-place approximations to the area under the curve f(x) = 1/x above the interval [1,3] on the x-axis. 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.
8. Modify the commands in your helper application file that compute left-hand sums so they will instead calculate right-hand sums for input values a, b, and n. Check that your new commands give right-hand sums that match those of our applet for each number of rectangles n given in the applet. Use the function f(x) = (x² + 5)/6.
9. Use the right-hand sum commands to calculate ten-decimal-place approximations to the area under the curve f(x) = 1/x above the interval [1,3] on the x-axis. 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.