Page 253 - DMTH401_REAL ANALYSIS

Unit 20: The Riemann Integration

Notes
1 1 1 3 1
Solution: For partition P , n = 5, x = 0, x = , x = , x = , x = , x = 1 and so  x = ,
1
1 0 1 2 1 4 4
4 3 2 4 4
1 1 1 1
 x = ,  x = ,  x = ,  x = .
2
5
4
3
12 6 4 4
5 4 3 7
Further M = f(x ) & m = f(x ) for i = 1, 2, 3, 4, 5 and therefore M = , M = , M = , M = ,
i i i i-1 1 2 3 4
4 3 2 4
5 4 3 7 5 25
M = 2, m = 1, m = , m = , m = , m = . We have L(P ,f) = å m  x = and
i
i
1
5 1 2 3 4 5 18
=
4 3 2 4 i 1
5 29 17 19
U(P ,f) = å M A x, = . Similarly, L(P ,f) = , and U(P ,f) = . Hence L(P ,f) L(P ,f)£
1 1 2 2 1 2
i 1 18 12 12
=
and U(P ,f) U(P ,f).£ 1
2
b b
Theorem 2: f(x) dx £  f(x) dx.

a a
Proof: If P & P , are two partitions of [a,b] and P = P U P, is their common refinement, then using
1 2 1
Theorem 1, we have L(P ,f) L(P,f) U(P,f) U(P ,f)£ £ £ 1 and
1
L(P ,f) L(P,f) U(P,f) U(P ,f).
£
£
£
2 2
Therefore, L(P ,f) U(P ,f).£
1 2
Keeping P fixed and taking l.u.b. over all P , we get
2 1
b
 f(x) dx £ U(P ,f)
2
a
Now taking g.l.b. over all P , we obtain
2
b b
 f(x) dx  £ f(x) dx
a a
This proves the result.
In Theorem 1, we have compared the lower and upper sums for a partition P with those for a
1
finer partition P . Next theorem, which we state without proof, gives the estimate of the difference
2
of these sums.
Theorem 3: If a refinement P, of P contains p more points and f(x) £ k, for all x [a,b],Î then
1
L(P ,f) L(P ,f) L(P ,f) 2pk ,
£
£
d
+
1
2
1
and U(P ,f) U(P ,f) U(P ,f) 2p k ,  - d where d is the norm of P .
1 2 1 1
This theorem helps us in proving Darboux's theorem which will enable us to derive conditions
of integrability. Firstly, we give Darboux's Theorem.
Theorem 4: 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
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