Theorem Cauchy-Inequality for IntegrationDefinite Integrals
Definition Let be a continuous function defined on divide the interval by the points
from to into subintervals. (not necessarily equal width) such that when , the length of each subinterval will tend to zero.
In the ith subinterval choose for . If exists and is independent of the particular choice of and , then we have
Remark For equal width, i.e. divide into equal subintervals of length, i.e. ,
we have .
Choose and
OR
Example Evaluate
Example =
Example Using a definite integral, evaluate
(a)
(b)
AL95II(a) Evaluate , where .
(b) By considering a suitable definite integral, evaluate
AL83II-1 Evaluate (a) ,
(b) , [ Hint: Put .]
(c)
Example* =
Properties of Definite Integrals
P1 The value of the definite integral of a given function is a real number, depending on its lower
and upper limits only, and is independent of the choice of the variable of integration, i.e.
.
P2
P3
P4 Let , then
Example (a) (b) (c)
P5* Comparison of two integrals
If , then
Example , for all ;
hence .
Example Prove that (a) .
(b)
ExampleIn Figure, is tangent to the curve at , where .
By considering the area of , show that
Hence show that for any integer .
P6 Rules of Integration
If are continuous function on then
(a) for some constant k.
(b) .
P7* (a) . a : any real constant.
(b) .
(c)
(d)
Proof(a)
Example Evaluate (a) (b)
Exercise 7C
5. By proving that
evaluate (a) (b)
6 (a) Show that
(b) Using (a), or otherwise, evaluate the following integrals:
(i) (iii)
Remind are odd functions.
are even functions.
Graph of an odd functionGraph of an even function
P8 (i) If (Even Function)
then
(ii) If (Odd Function)
then
Proof
ExampleEvaluate (a) (b)
Example Prove that (a) (b)
Definition Let be a subset of , and be a real-valued function defined on . is called a periodic function if and only if there is a positive real number T such that , for . The number T is called the period.
P9 If is periodic function, with period i.e.
(a)
(b)
(c)
(d) for
Proof
Theorem Cauchy-Inequality for Integration
If , are continuous function on , then
Proof
Example
Date: 2016-04-22; view: 964
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