Graphics Reference
In-Depth Information
C
B
A
a
c h
c + h
b
A
on [
a
,
c
),
f
(
c
)
C
,
f
(
x
)
f
(
x
)
B
on (
c
,
b
],
by using theequation
f
f
f
f
.
Start working with
A
B
C
.
15 Let
a
x
x
x
...
x
b
, and a real function
f
be
defined on [
a
,
b
] such that
f
(
x
)
A
, for
x
x
x
,
i
1, 2, . . .,
n
; and
f
(
x
,
i
0, 1, 2, . . .,
n
.
Such a function is called a
step function
.
Use qn 14 to prove that if this step function has an integral
compatible with properties (i) and (ii) before qn 12, then its value is
)
B
(
x
x
)
A
.
Wenow adopt this rsult as the
definition
of the integral of a step
function.
Lower integral and upper integral
R
16 Consider a function
f
:[
a
,
b
]
bea step
function such that
s
(
x
)
f
(
x
) for all values of
x
. Such a step
function is called a
lower step function
for
f
.
[
m
,
M
]. Let
s
:[
a
,
b
]
Wecall
s
,a
lower sum
for
f
.
Why must
m
(
b
a
) bea lowr sum for
f
?
Why must every lower sum be
M
(
b
a
)?
Since, for the given bounded function
f
, the set of all lower sums is
