Inequalities
Sometimes you are
asked to prove inequalities involving definite integrals or to estimate the
upper and lower bounds of a definite integral, where the exact value of the
definite integral is difficult to find. Under these circumstances, we use the
following results:.
(i) 
Equality sign holds
where f (x) is entirely of the same sign on [a, b]
Example.1
Estimate the absolute value of the integral
.
Solution
Since |sinx| £ 1 for x ≥ 10, the inequality 
< 10-8
is fulfilled.
Therefore 
< (19 – 10) 10-8
< 10-7 (the true value of the integral » -10-8).
(ii) 
Example.1
Prove that 
can’t
exceed 
.
Solution
£ 
.
(iii) If f (x) ≥ g(x) on [a, b] , then 
. In particular, if f
(x) ≥ 0, then 
.
Example.1
If f (x) is a continuous function such that
f (x) ≥ 0 " x ∈ [2, 10] and 
Solution
f(x) is above the
x-axis or on the x-axis for all x ∈
[2, 10]. If f (x) is greater than zero for any sub interval of [4, 8], then 
f (x) d x must be
greater than zero. Butf (x) dx = 0
&⇒ f (x) = 0 " x ∈ [4, 8]
&⇒ f (6) = 0
(iv). For a given
function f (x) continuous on [a, b] if you are able to find two
continuous function f1(x) and f 2(x) on [a, b]
such that f1(x) £
f(x) £ f2(x) "x ∈ [a, b], then .
.
Example.1
Prove that 
Solution
Since 4 – x2
≥ 4 – x2 –
x3 ≥ 4 – 2x2 >
0 " x ∈ [0, 1]
.
(v). If m and M are the global minimum and global maximum
of f (x) respectively in [a, b] then m (b – a) 
Example.1
Prove
that 4 
.
Solution
Since the function f (x) = 
increases
monotonically on the interval [1, 3], m = 2, M = 
, b – a = 2.
Hence, 2.2
&⇒
4 .
