In Part 1 we used rectangles whose heights matched the height of the curve at the left or right endpoint of the subinterval which forms the base of the rectangle. We can improve the accuracy of our area estimate by instead using rectangles whose heights match the curve at the midpoint of the base subinterval.

Use the applet below to experiment with changing the number of midpoint rectangles and changing the function generating the curve. Be sure to click the "horizontal" button first or you won't see midpoint rectangles.

In the applet above, click the "tangent" button and experiment again with different numbers of subintervals n. Note that the rectangles become trapezoids, but not like those we used in Part 2. These trapezoids have tops that are tangent to the curve at the midpoint of the subinterval.

For a fixed number of subintervals, alternatively click "horizontal" and "tangent." You should see that the area approximation doen't change. We will attempt to find out why in step 1 below.

1. For the quarter circle graph, show n = 2 midpoint rectangles. On a separate piece of paper draw the second of the two rectangles. Label the vertices of the rectangle in your diagram as A, B, C, and D starting by labeling the top left vertex as A and moving clockwise. Label the midpoint on the top boundary of the rectangle as M.

   Now click the "tangent" button to show two trapezoids and again concentrate on the second one. On your diagram, add the tangent line passing through point M. Label the left end of the tangent line R and the right T. Shade in the right triangles AMR and BMT with pencil on your drawing. How do we know that these two triangles have the same area?

   Use this fact to argue that the area of the midpoint rectangle (ABCD) is exactly the same as the area of the trapezoid (RTCD). Argue that this will always be true regardless of the function graph being used and thus the sum of the areas of the trapezoids will alway equal the sum of the areas of the midpoint rectangles.

2. For the parabolic function y = (x² + 5)/6, is the midpoint sum an upper bound for the area under the curve, or is it a lower bound, or is it neither? What if we ask the same question about the function y = sqrt(25 - x²)? (Hint: Here is where you need your result from step 1.)
3. What property must a function have to be sure the midpoint sum gives an upper bound for the true area under the curve?, a lower bound? Be careful, it doesn't depend on the function being increasing or decreasing.
4. Define the function f(x) = (x² + 5)/6 in your helper application. Then calculate the midpoint sum with n = 5 rectangles and compare your result to that given in the applet. Note that the heights of the five rectangles are f(0.5), f(1.5), f(2.5), f(3.5) and f(4.5) and all the bases are 1 unit long.
5. Suppose we want to approximate the area under a function f(x) above the interval [a, b] on the x-axis using a midpoint sum with n rectangles, all with equal bases. Then each base must have length delx = (b - a)/n. The base of the first rectangle extends from a to a + delx along the x-axis and its midpoint is a + (1/2)*delx. The base of the second rectangle extends from a + delx to a + 2*delx along the x-axis and its midpoint is a + (3/2)*delx.

   In terms of parameters a and delx, the i -th rectangle's base extends between what two x values? What is a formula for its midpoint? What is the formula for its area? (Your formulas will depend on parameters a, delx, and i.)

6. In the helper application file that accompanies this module, you will find commands based upon the results of part 5 above that will calculate a midpoint sum for input values a, b, and n. Check that these commands give 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 midpoint 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 later in this module.