Graphics Reference
In-Depth Information
R
f
:[
a
,
b
]
0,
f
has a
lower step function
s
with lower sum
and an upper step function
S
with an upper sum
has integral
F
, and, following qn 23, given
, such that
.
26 If, for
k
0, thefunction
k
·
f
is defined by
k
·
f
:
x
k
·
f
(
x
),
show that
k
·
s
is a lower step function and
k
·
S
is an upper step
function for
k
·
f
and deduce from qn 24 that
k
·
f
is integrable, and
has integral
k
·
F
.
f
is defined by
27 If thefunction
f
:
x
f
(
x
),
show that
s
is an upper step
function for
f
, and deduce from qn 24 that
f
is integrable and
has integral
F
.
S
is a lower step function and
28 From qns 26 and 27 extend the result of qn 26 to show that the
result in that question holds for any real value of
k
.
has
integral
G
and, following qn 23, given
0,
g
has a lower step function
s
We suppose further that the integrable function
g
:[
a
,
b
]
R
and an upper step function
S
with lower sum
with upper sum
,
such that
.
29 If thefunction
f
g
:[
a
,
b
]
R
is defined by
f
g
:
x
f
(
x
)
g
(
x
),
show that
s
s
is a lower step function for
f
g
and that
S
S
is
an upper step function for
f
g
, and that the difference between the
upper and lower sums for these step functions is (
)
(
).
Deduce that
f
g
is integrable and has integral
F
G
.
Putting together the results of qns 28 and 29, we can show that, if
f
:[
a
,
b
]
R
R
is integrable
with integral
G
, then the function
k
·
f
l
·
g
is integrable with integral
k
·
F
l
·
G
. This makes the obtaining of an integral a linear function of
thefamily of integrablefunctions on thedomain [
a
,
b
].
is integrable with integral
F
and
g
:[
a
,
b
]
30 Use qns 28 and 29 with qn 4 to determine the value of
f
when:
(i)
f
(
x
)
x
x
;
(ii)
f
(
x
)
2
x
3
x
4
x
;
(iii)
f
(
x
)
c
c
x
c
x
...
c
x
.
