The Definite Integrals
Definite Integrals Advanced Level Pure Mathematics
Definite Integrals 1
Properties of Definite Integrals 3
Integration by Parts 4
Continuity and Differentiability of a Definite Integral 5
Improper Integrals 6
Definite 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 =
*AL95II (a) Evaluate , where .
(b) By considering a suitable definite integral, evaluate
AL83II-1 Evaluate (a) ,
(b) , [ Hint: Put .]
*(c)
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
P5* Comparison of two integrals
If , then
Example , for all ;
hence .
Example Prove that (a) .
(b)
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)
Exercise 7C
1. * By proving that (7c5)
evaluate (a) (b)
2 (a) Show that (7c6)
(b) Using (a), or otherwise, evaluate the following integrals:
(i) (iii)
Remind are odd functions.
are even functions.
Graph of an odd function Graph of an even function
P8 (i) If (Even Function)
then
(ii) If (Odd Function)
then
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
Theorem Cauchy-Inequality for Integration
If , are continuous function on , then
Theorem Triangle Inequality for Integration
Integration by Parts
Theorem Integration by Parts
Let and be two functions in . If and are continuous on , then
or
* AL84II-1 (a) For any non-negative integer k, let .
Express in terms of .
Hence, or otherwise, evaluate .
(b) For any non-negative integers and , let
(i) Show that if , then
= .
(ii) Evaluate for .
(iii) Show that if , then
= .
(iv) Evaluate .
AL94II-11 For any non-negative integer , let
(a) (i) Show that .
[ Note: You may assume without proof that for . ]
(ii) Using (i), or otherwise, evaluate .
(iii) Show that for .
(b) For , let .
(i) Using (a)(iii), or otherwise, express in terms of .
(ii) Evaluate .
Continuity and Differentiability of a Definite Integral
Theorem Mean Value Theorem for Integral
If is continuous on [a,b] then there exists some c in [a,b] and
Theorem Continuity of definite Integral
If is continuous on and let then is continuous at each point x in .
Theorem * Fundamental Theorem of Calculus
Let be continuous on [a,b] and . Then ,
Remark :
Example Let be a function which is twice-differentiable and with continuous second derivative. Show that , .
* AL90II-5 (a) Evaluate , where is continuous and is a positive integer.
(b) If , find
AL97II-5(b) Evaluate .
AL98II-2 Let be a continuous periodic function with period .
(a) Evaluate
(b) Using (a), or otherwise, show that for all .
Improper Integrals
Definition A definite integral is called improper integral if the interval [a,b] of integration is infinite, or if is not defined or not bounded at one or more points in [a,b].
Example , , , are improper integral.
Definition (a) is defined as
(b) is defined as
(c) is defined as for any real number .
( Or )
(d) If is continuous except at a finite number of points, say where
, then is defined to be
for any such that .
Definition The improper integral is said to be Convergent or Divergent according to the improper integral exists or not.
Theorem Let and be two real-valued function continuous for . If then the fact that diverges implies diverges and the fact that
converges implies that converges.
Example Determine whether the improper integrals are convergent or divergent.
a) (p.109 7-45c)
b) (p.112 7-50 b)
c) (p.112 7-50 c)
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