Page 271 - DMTH401_REAL ANALYSIS

Unit 21: Properties of Integrals

such that Notes
U(P,f) – L(P,f) £Î
Let P = {x , x , x , …, x }.
0 1 2 n
¢
¢
Let M and m denote the supremum and infimum of f and M and m denote the supremum
i i i i
and infimum of f in [x , x ].
i-1 i
You can easily check that
'
'
M - m  M - m .
i i i i
n n
¢
¢
D
This implies that å (M - m ) A x  å (M - m ) x ,
i i i 1 1 i
=
=
i 1 i 1
-
i.e., (P, f ) L P, f- ( ) £ U(P,f) L(P,f) <Î ,
U
This shows that f is integrable on [a,b].
Note that the inequality established in Property 2 may be thought of as a Integrability and
differentiability generalization of the well-known triangle inequality
a b £ a + b
+
In other words, the absolute value of the limit of a sum never exceeds the limit of the sum of the
absolute values.
b
You know that in the integral f(x) dx, the lower limit a is less than the upper limit b. It is not
ò
a
always necessary. In fact the next property deals with the integral in which the lower limit a may
be less than or equal to or greater than the upper limit b.
For that, we have the following definition:
b
Definition 1: Let f be integrable on [a,b], that is, f(x) dx exists when b > a. Then
ò
a
b
ò f(x) dx = 0, if a = b
a

a
= ò - f(x) dx, if a > b.
b
Now have the following property.

b
-
Î
Property 3: If a function f is integrable in [a,b] and f(x) £ k " x [a,b], then f(x) dx £ k b a .
ò
a
Proof: There are only three possibilities namely either a < b or a > b or a = b. We discuss the cases
as follows:
Case (i): a < b
Since f(x) £ k " x [a,b], therefore
Î

– k £ f(x) £ k " x [a,b]
Î

LOVELY PROFESSIONAL UNIVERSITY 265