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# 8.5 Change of Scale

8.42   Definition (Stretch of a function.) Let be an interval in , let , and let . We define a new function by

If , then , so is defined.

The graph of is obtained by stretching the graph of by a factor of in the horizontal direction, and leaving it unstretched in the vertical direction. (If the stretch is actually a shrink.) I will call the stretch of by .

8.43   Theorem (Change of scale for integrals.) Let be an interval in and let . Let and let be the stretch of by . If is integrable on then is integrable on and , i.e.,
 (8.44)

Proof: Suppose is integrable on . Let be an arbitrary partition-sample sequence for . If

let

Then is a partition-sample sequence for , so
. Now

so

This shows that is integrable on , and .

Remark: The notation is not a standard notation for the stretch of a function, and I will not use this notation in the future. I will usually use the change of scale theorem in the form of equation (8.44), or in the equivalent form

 (8.45)

8.46   Exercise. A Explain why formula (8.45) is equivalent to formula (8.44).

8.47   Example. We define to be the area of the unit circle. Since the unit circle is carried to itself by reflections about the horizontal and vertical axes, we have

Since points in the unit circle satisfy or , we get

We will use this result to calculate the area of a circle of radius . The points on the circle with radius and center 0 satisfy , and by the same symmetry arguments we just gave

By the change of scale theorem

The formulas

or more generally

are worth remembering. Actually, these are cases of a formula you already know, since they say that the area of a circle of radius is .

8.48   Exercise. A Let be positive numbers and let be the set of points inside the ellipse whose equation is

Calculate the area of .

8.49   Exercise. The figure shows the graph of a function .
Let functions , , , , and be defined by
a)
.
b)
.
c)
.
d)
.
e)
.
Sketch the graphs of ,,, , and on different axes. Use the same scale for all of the graphs, and use the same scale on the -axis and the -axis,

8.50   Exercise. A The value of is (approximately). Use this fact to calculate approximate values for

where . Find numerical values for both of these integrals when .

Next: 8.6 Integrals and Area Up: 8. Integrable Functions Previous: 8.4 The Ruler Function   Index
Ray Mayer 2007-09-07