Functions Defined by Integrals, Part 1

Functions Defined by Integrals

Part 1: The Function F(x) and its Derivative

Many useful functions are naturally described by integration of known functions. Sometimes these newly defined functions turn out to be functions we already know in some other context -- sometimes they are truly new. In this part we define a function F(x) by integrating a given continuous function f(x), and then we study the derivative of the new function F(x).

For our initial example, we start with the function

whose graph is a quarter-circle of radius 5:

We define a new function F(x) by setting its value at a positive number x to be the integral of f from 0 to x:

Note that, since x is being used to represent the independent variable for the function F, a different variable t is used for the variable of integration.

What does this function definition mean? Since the definite integral of a positive function f represents area under the graph of f, we may think of F(x) as the area under the graph of f(t) between t = 0 and t = x. We illustrate this in the following figure.

To see the construction of the function F dynamically, click on one of the following animations. But first, please read these instructions.

Each animation appears in a new browser window. To return to this page, close the animation window.
To start an animation, click anywhere in the window. You may do this as many times as you like.
If your mouse has a right button, a right click will bring up options for pausing the animation, moving a frame at a time, and others.
If your computer is connected directly to the Internet, select the smoothest animation. If you are using dial-up access, note the file sizes -- download time is proportional to file size.
The horizontal scales for the graphs of f and F are the same, from x = 0 to x = 5. But note that the vertical scales are different.

F(x) animation options: Least smooth (93K), smoother (316K), smoothest (762K).

Use the definition of F(x) in your worksheet to make your own graph of F(x). Examine the graph, and explain why it has the form it does.
Make a table of values of F(x) for x ranging from 0 to 5. (Don't limit yourself only to integer values of x.) Make sure the values you get are consistent with your graph of F(x).

We turn our attention now first to estimating, then to calculating, the derivative of F(x). For purposes of estimation, we will calculate difference quotients

with the difference = 0.001. (Any reasonably small value of delta-x would do -- you are welcome to experiment with other values in your worksheet.)

Calculate the approximate derivative of F (difference quotient) and the corresponding value of f at each of the x values specified in your worksheet. Fill in the table in your worksheet as you go.

Your results in step 3 should strongly suggest a connection between F'(x) and f(x). However, a handful of approximately equal pairs of numbers is not enough to establish equality. We consider now why that apparent equality might actually be true.

First we consider the change in function values of F when x is incremented by delta-x :

by the definition of F and a property of definite integrals. Thus, the change in F is the area under the graph of f between x and x + . [Note: We are using implicitly the fact that f(x) is nonnegative when we refer to area. However, the conclusion remains correct for any continuous function if we use the concept of "signed area."]

What property of integrals is referred to in the preceding paragraph?
Describe in geometric terms why the change in F is the indicated area. [Hint: Interpret the definition of F in terms of area.]

We illustrate the area interpretation of the change in F in the next pair of figures, using x = 3 and = 0.001. The left figure shows the area under the graph of f between 3 and 3.001 shaded in green. Because the interval is so small, the shaded region looks rather like a line. In the right figure, we have left the vertical scale the same but have greatly expanded the horizontal scale. Again, the area under the graph of f between 3 and 3.001 is shaded in green.

Why does the graph of f on [3, 3.001] appear to be a horizontal line?
Give a quick estimate of the shaded area.
For an arbitrary x and an arbitrary (small) , give a quick estimate of the area under the graph of f over the interval [x, x + ].
Explain the estimate for the difference quotient:
Observe that your estimate for the difference quotient
- improves as gets smaller, and
- is independent of .
Explain the following calculation of the instantaneous rate of change of F:
For our specific example,

use your helper application to confirm the formula in step 10. That is, ask the computer algebra system for a symbolic formula for F(x). Then ask for the derivative of that formula, and confirm that the result is equivalent to the formula for f(x).
(optional) Explain why F is the solution of the initial value problem