Graphics Reference
In-Depth Information
R
Definition
qns 16, 18
If
s
:[
a
,
b
]
is a step function and
f
:[
a
,
b
]
R is a function,
s
is called a
lower step
function
for
f
when
s
(
x
)
f
(
x
) for all values of
x
,
and
s
is called an
upper step function
for
f
when
f
(
x
)
s
(
x
) for all values of
x
.
When
s
is a lower step function, its integral is
called a
lower sum
for
f
. When
s
is an upper step
function, its integral is called an
upper sum
for
f
.
Theorem
qns 16, 18
Any bounded function
f
:[
a
,
b
]
[
m
,
M
] has an
upper sum
M
(
b
a
) and a lower sum
m
(
b
a
).
Theorem
qn 20
Each lower sum of a bounded function
f
:
[
a
,
b
]
R
is less than or equal to each upper
sum.
The Riemann integral
Definition
qns 16, 18
The supremum of the lower sums of a bounded
function
f
:[
a
,
b
]
R is called the
lower integral
of
f
and denoted by
f
.
Theinfimum of theuppr sums of
f
is called the
upper integral
of
f
and denoted by
f
.
R
Theorem
qn 21
For a given bounded function
f
:[
a
,
b
]
,
each lower sum
the lower integral
each lower sum
the upper integral
each lower sum
each upper sum.
Definition
qn 22
A bounded function
f
:[
a
,
b
]
is said to be
Riemann integrable
when its upper integral is
equal to its lower integral. Their common value
is called the
Riemann integral
of thefunction
and denoted by
R
f
.
Theorem
qns 23, 24
A function
f
:[
a
,
b
]
is Riemann integrable if
and only if, given
0, there exists an upper
sum
R
and a lower sum
, such that
.
Theorem
qns 7, 8, 10
Every function which is monotonic on a closed
interval is integrable.
Theorems on integrability
While obtaining the basic algebraic properties of integrals (in qns
26
—
31) it will beusful to havenotation availablethat will not haveto
be repeatedly defined. We suppose that the integrable function
