Page 267 - DMTH401_REAL ANALYSIS

Unit 20: The Riemann Integration

b Notes
denoted by f(x) dx. Also in this section, it has been shown that in passing from a partition

a
PI to a finer partition P2, the upper sum does not increase and the lower sum does not
decrease. Further, you have seen that the lower integrable of a function is less than or
equal to the upper integral. Further condition of integrability has been derived with the
help of which the integrability of a function can be decided without finding the upper and
lower integrals. Using the condition of integrability, it has been shown that a function f is
integrable on [a,b] if it is continuous or it has a finite number of points of discontinuities
or the set of points of discontinuities have finite number of limit points. Also you have
seen that a monotonic function is integrable. As in the case of continuous and derivable
functions, the sum, difference, product and quotient of integrable functions is integrable.
Riemann sum S(P,f) of a function f for a partition P has been defined and you have been
b
shown that lim S(P,f) exists if and only if f Î R [a,b] and f(x) dx = lim S(P,f). Using this

P  0 a P  0
idea a number of problems can be solved.
20.6 Keywords

Partition: Let [a,b] be a given interval. By a partition P of [a,b] we mean a finite set of points
{x , x , ...., x,}, where
0 l
a = x , < x <… < x <x = b.
0 1 n-1 n
We write x = x – x , (i=l, 2, ..., n). So x is the length of the ith sub-interval given by the
i i i-1 i
partition P.

Norm of a Partition: Norm of a partition P, denoted by P , is defined by P = max Ax,. Namely,
i
t £ £ n
the norm of P is the length of largest sub-interval of [a, b] induced by P. Norm of P is also
denoted by (P).
Darboux's Theorem: If f: [a,b]  R is a bounded function, then to every Î > 0, there corresponds
d > 0 such that
b
(i) U(P,f) <  f(x) dx + Î
a

b
(ii) L(P,f) <  f(x) dx - Î
a

20.7 Review Questions

1. Find the upper and lower Riemariri integrals of the function f defined in [a, b] as follows

ì 1 when x is ratinal
f(x) = í
î - 1 when x is irrational
2. Show that the function f where f(x) = x[x] is integrable in [0, 2].

n
3. Show that the function f defined in LO, 21 such that f(x) = 0, when x = or
n 1
+
n 1
+
¼
(n = 1,2,3, ), and f(x) = 1, elsewhere, is integrable.
n

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