Go to CCP Homepage Go to Materials Page Go to PostCALC Materials Go to Table of Contents
Go Back One Page Go Forward One Page

Fourier Approximations and Music

Part 3: Calculus Background

In this part of the module we will begin by assuming that the functions under investigation have period 2 pi. As a first step in the process of breaking down complex periodic functions into simple sine and cosine functions we investigate definite integrals of the forms given below:


Here m and n are integers. We will consider both the case where m is not equal to n and the case where the two integers are equal.

1. Evaluate the following integrals by hand. (Be careful in the cases where m = 0.)

      (a)       (b)       (c)       (d)

2. We now investigate integrals of the form

.
For each of the following functions, estimate the value of the integral by studying the graphs produced on the computer algebra worksheet.

      (a)             (b)
      (c)             (d)

3. What can you conclude about


where m and n are integers?

4. Verify the conclusions you reached in answering Question 3 by evaluating appropriate integrals on your worksheet for a range of values of m and n.

5. Explain your answer to Question 3 in light of what you know about odd and even functions.

6. We now turn to integrals of the form


by looking at the graphs of the integrands.

      (a) What general conjecture can you make about

       by looking at the graph of y = sin(t) sin(2t) for t between -pi and pi?

       (b) Study the graph of y = sin(5t) sin(3t). Does this change the conjecture you made in (a)?

7.    (a) Verify the conjecture you made in the last question by evaluating the integrals on your worksheet
       with various values for m and n where m is not equal to n.

       (b) We now look at a general formula for integrals of the form

       (c) Plot the graphs of at least three functions of the form f(t) = sin(mt) sin(nt), where m = n.
       Make a conjecture about the integrals in this case.

       (d) Verify your conjecture made in (c) by using your computer algebra system to evaluate the
       integrals of the functions you graphed and by analyzing the general formula.

8. Make a final conjecture about integrals of the form

9. Investigate integrals of the form

using the same approach as was used in Questions 6 and 7.

Go to CCP Homepage Go to Materials Page Go to PostCALC Materials Go to Table of Contents
Go Back One Page Go Forward One Page

modules at math.duke.edu Copyright CCP and the author(s), 1998