Graphics Reference
In-Depth Information
44 Thefunction
f
is continuous on [0, 1] by qn 6.49 and is thus integrable.
s
(
x
)
1 is an upper step
function for
g
. Consider step functions on [0, 1/
n
] and [1/
n
, 1]. The
function is continuous on [1/
n
, 1] and therefore there are arbitrarily
close upper and lower sums. On [0, 1/
n
] the difference between upper
and lower sums is 2/
n
which may bearbitrarily small. So thefunction
is integrable by qn 24.
1 is a lower step function and
S
(
x
)
45 See qns 16 and 18. Since
f
is continuous, (
b
a
)
f
is continuous. The
minimum valueof this function is
m
(
b
a
) and themaximum valueof
this function is
M
(
b
a
). Since the integral lies between these values,
or at one of them, the integral is equal to (
b
a
)
f
(
c
) for some
c
[
a
,
b
]
by the Intermediate Value Theorem.
Dividethersult of qn 43(ii) by (
b
a
).
46 Since
f
is integrable on [
a
,
b
], there exist step functions
s
and
S
giving
upper and lower sums which arearbitrarily clos. If
c
and
d
were not
part of the subdivisions for these step functions, introduce these points
into the subdivision for each step function. Then the difference between
the upper and lower sums on [
c
,
d
]
the difference between the upper
and lower sums on [
a
,
b
], so
f
is integrable on [
c
,
d
].
47 If
f
is a step function the result is obvious. To obtain the result for any
integrable
f
, apply this result to upper and lower step functions for
f
.
f
48 To retain the equation for qn 47, we must define
0 and
f
f
.
49 Thefunction
f
is monotonic.
0
when 0
x
1,
x
x
F
(
x
)
1
when 1
2,
2
x
x
3
when 2
3.
F
is continuous at
x
1 and 2, by considering limits from above
and below at each of these points.
50 Since
f
is integrable,
f
is bounded. Let
m
f
(
x
)
M
.
F
(
c
h
)
F
(
c
)
f
, so for positive
h
,
mh
F
(
c
h
)
F
(
c
)
Mh
.
(
F
(
c
h
)
F
(
c
))
F
(
c
h
)
F
(
c
).
Thus lim
0, so lim
Likewise for the limit from below. Then use qn 6.89, continuity by
limits.
L
max
m
,
M
.
51
(i) By the Mean Value Theorem applied to
F
.
(iii) Every summation of this kind lies between an upper sum and
