Page 306 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Therefore,
1 p p p 1
1 £  £ i.e. £ £ - ,
4 6 6 6 k 2
1 -
4
1
p 2 1 p 1
and; so, £ ò dx £ .
6 0 é ë (1 x )(1 k x )ù û 6 1 - k 2
2
2
2
-
-
4
q
sinx 2
Example: Prove that ò dx £ , if q > p > 0.
p x p
1
Solution: Let f(x) = sin x, (x) = ,x [p,q]. Being continuous, these functions are integrable in


x
[p, q]. By Bonnet form of second mean value theorem, there is a point [p, q] such that
q 
ò f(x) (x) dx =  (p) f(x) dx
ò

p a
q sinx 1  1
i.e., ò dx = ò sinx dx = (cosp cos ).

-
p x p p P
q
sinx 1 2

Hence ò dx £ é ë cosp + cos ù £
û
p x P P
Self Assessment
Fill in the blanks:
1. Let f : [a, b]  R be a continuous function. Then there exists c  [a, b] such that .......................
2. Since f is continuous in [a, b], it attains its bounds and it also attains every value between
the ...........................
3. The geometrical interpretation of the theorem is that for a ....................................... function
f, the area between f, the lines x = a, x = b and the x-axis can be taken as the area of a
rectangle having one side of length (b-a) and the other f(c) for some c  [a, b].
4. If f and g are differentiable functions Qn [a,b] such that the derivatives f' and g' are both
................................ on [a, b], then
b b
ò f(x) g' dx [f(b) g(b) f(a) g(b)]- ò f'(x) g(x) dx.
-
=
a a
5. Let f and g be any two functions integrable in a,b and g be .............................. then there
exists c  a,b such that

b c b
+
ò
ò f(x) g(x) dx = g(a) f(x) dx g(b) f(x) dx
ò
a a c

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