Better approximations of the area under a curve can be found by breaking the area up into trapezoids instead of rectangles. The applet below adds up the areas of a set of trapezoids to approximate the area under the graph of a function. The functions available are the same as those for the previous applet.

Experiment with changing the number of trapezoids and changing the function generating the curve. First try n = 2 subintervals to get a good look at two large trapezoids.

1. For the parabolic function y = (x² + 5)/6, is the trapezoidal 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)? Be sure to first look at trapezoids for small n values. Otherwise the fit may be hard to see.
2. What property must a function have to be sure the trapezoidal 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.
3. Define the function y = (x² + 5)/6 in your helper application. Then calculate the trapezoidal sum with n = 5 trapezoids and compare your result to that given in the applet. Note that the area of a trapezoid can be found by multiplying the average height times the width of the base. Thus the first trapezoid in your sum has area (f(0) + f(1))/2 * 1.0.
4. Suppose we want to approximate the area under a function f(x) above the interval [a, b] on the x-axis using a trapezoidal sum with n trapezoids, all with equal bases. Then each base must have length delx = (b - a)/n. The base of the first trapezoid extends from a to a + delx along the x-axis. The base of the second trapezoid extends from a + delx to a + 2*delx along the x-axis.

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

5. In the helper application file that accompanies this module, you will find commands based on the results of step 4 above that will calculate a trapezoidal sum for input values a, b, and n. Check that these commands give sums that match those of our applet for each number of trapezoids n given in the applet. Use the function f(x) = (x² + 5)/6.
6. Use the trapezoidal 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.
7. Go back to Part 1 and calculate the left-hand sum and the right-hand sum for n = 5 for the function f(x) = (x² + 5)/6. Find the average of these values: (L + R)/2. Compare this result to the trapezoidal sum for n = 5. You should find the results are the same. Does this property always hold? Experiment with different functions and numbers of subintervals.
8. In order to verify the property we discovered in the previous step, do a little algebra. Suppose the interval [a, b] is divided into subintervals of equal length delx at the division points x₀, x₁, x₂, ... , x_n. Then the first term in the left-hand sum is f(x₀)*delx. Likewise, the first term in the right-hand sum is f(x₁)*delx. Now substitute these two first terms into (L + R)/2 and show that this expression is algebraically equivalent to the first term in the trapezoidal sum.

   You will find a similar result if you average the second term in the L sum with the second term in the R sum. Argue that this generalizes to all terms so that (L + R)/2 will be algebraically equal to the trapezoidal sum.