Next: 8.6 Integrals and Area
Up: 8. Integrable Functions
Previous: 8.4 The Ruler Function
Index
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 .
Proof: Suppose is integrable on . Let
be an
arbitrary partitionsample sequence for . If
let
Then
is a
partitionsample 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
20070907