Page 251 - DMTH401_REAL ANALYSIS

Unit 20: The Riemann Integration

Notes
n n
-
U(P,f) = å M  x = å  x = b a
i
i
i
=
=
i 1 i 1
n
U(P,f) = å m  x = 0
i
i
i 1
=
Therefore U(P,f) = b – a and L(P,f) = 0 for every, partition P of [a,b]. Hence
b
 f(x) dx = g.l.b. {U(P,f): P is a partition of [a,b]]
a
= g.l.b. {b – a] = b – a.
b
 f(x)dx = l.u.b. {L(P,f): P is a partition of [a,b]]
a
= l.u.b. {0] = 0.
Example: Let f be a constant function defined in [a,b]. Let f(x) = k  x Î [a,b]. Find the
upper and lower integrals of f.
Solution: With the same notation as in example 1, M = k and m = k  i.
i i
n n
-
U(P,f) = å M A x = å A x = k(b a)
i
i
i
i 1 i 1
=
=
n n
and L(P,f) = å m A x = å A xi = k(b a)
-
i
i
=
=
i 1 i 1
Therefore U(P,f) = k(b – a) and L(P,f) = k (b – a) for every partition P of [a,b].
b b
Consequently f(x)dx = k(b a) and f(x)dx = k(b a)
-
-


a a
Now try the following exercise.
Exercise
Find the upper and lower Riemann 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
You have seen that sometimes the upper and lower integrals are equal (as in Example) and
sometimes they are not equal (as in Example). Whenever they are equal, the function is said to
be integrable. So integrability is defined as follows:
Definition 6: Riemann Integral
Let f: [a,b] – R be a bounded function. The function f is said to be Riemann integrable or simply
b b
integrable or R-integrable over [a,b] if f(x) dx =  f(x) dx and iff is Riemann integrable, we

a _
b
denote the common value by f(x) dx. This is called the Riemann integral a r simply the integral

a
off on [a, b].

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