Page 273 - DMTH401_REAL ANALYSIS

Unit 21: Properties of Integrals

Property 4: If f is integrable on [a,b] and c Î [a,b], then f is integrable on [a,c] and [c,b] and Notes
conversely. Further in either case

b c b
ò f(x) dx = ò f(x) dx + ò f(x) dx.
a a c
According to this property, if we split the interval over which we are integrating into two parts,
the value of the integral over the whole will be the sum of the two integrals over the subintervals.
This amounts to dividing the region whose area must be found into two separate parts while the
total area is the sum of the areas of the separate portions.

b
We now state a few more properties of the definite integral f(x) dx, these are:
ò
a
-
(i) ò f(x) dx = ò f(a x) dx.
n a
2a a a
-
(ii) ò f(x) dx = ò f(x) dx + ò f(2a x)dx.
0 0 0
ì a
a ï 2 f(x) dx if f is an even function
ò
(iii) ò f(x) dx = í 0
- a ï 0 iff is an old function.
î
na a
(iv) ò f(x) dx = n f(x) dx if f is periodic with period 'a' and n is a positive integer provided the
ò
0 0
integrals exist.

Self Assessment

Fill in the blanks:
b b
1. If f, is ...................... on [a, b] then f is also integrable on [a,b] and f(x) dx £ ò f(x) dx.
ò
a a
2. The inequality follows at once from Property 1 provided it is known that f is .........................

f
on [a, b]. Indeed, you know that - f £ £ f .
b
-
3. If a function f is integrable in [a, b] and .................................., then f(x) dx £ k b a .
ò
a
4. If f is integrable on [a, b] and ................., then f is integrable on [a, c] and [c, b] and conversely.
21.2 Summary

 Sum of two Riemann Stieltjes integrable functions is also Riemann Stieltjes integrable.
 Scalar product of a Riemann Stieltjes integrable function is also Riemann Stieltjes integrable.
 Modulus of a Riemann Stieltjes integrable function is also Riemann Stieltjes integrable.
 Square of a Riemann Stieltjes integrable function is also Riemann Stieltjes integrable.
 If a function is Riemann Stieltjes integrable on an interval, then it is also Riemann Stieltjes
integrable on any of its subinterval.

LOVELY PROFESSIONAL UNIVERSITY 267