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In-Depth Information
44 Determine whether the functions
f
and
g
defined below are
integrable on the interval [0, 1].
f
(
x
)
x
sin(1/
x
) when
x
0, and
f
(0)
0.
g
(
x
)
sin(1/
x
) when
x
0, and
g
(0)
0.
For
g
you can consider [0, 1]
[0,
]
[
, 1].
Mean Value Theorem for integrals
45 (
Cauchy
, 1823) If
f
is a continuous function on [
a
,
b
],
m
inf
f
(
x
)
a
x
b
and
M
sup
f
(
x
)
a
x
b
,
explain why
m
(
b
a
)
f
M
(
b
a
).
Deduce that, for some
c
f
f
(
c
) ยท (
b
a
).
This is called the
Mean Value Theorem for integrals
.
If
a
x
[
a
,
b
],
x
x
...
x
b
, and
x
x
(
b
a
)/
n
,
show that
f
(
x
)
/
n
f
(
c
)as
n
.
Integration on subintervals
46 If a function
f
is integrable on [
a
,
b
] and
a
c
d
b
, provethat
f
is integrable on [
c
,
d
].
47 If a function
f
is integrable on [
a
,
b
] and
a
c
b
, provethat
f
f
f
. We started using this as an intuitively desirable
property for integrals, but now that we have a formal definition of
what an integral is we need to prove that it is a formal consequence
of thedefinition.
48 Makea definition of
f
for any function
f
, which is compatible
with thersult of qn 46 in case
c
a
or
c
b
.
Define
f
in a way which will extend the validity of
the result in qn 47 whatever the order of the numbers
a
,
b
and
c
on
the number line.
f
in terms of
Summary
-
Properties of the Riemann integral
Theorem
qns 3, 4, 5
If
f
(
x
)
x
for
n
an integer
1, then
f
a
/(
n
1).
