DETAILED CALCULUS III NOTES.

LECTURE 1
IMPROPER INTEGRALS AND THE MEAN VALUE THEOREM OF INTEGRAL
CALCULUS

1.0 Introduction

Dear learner, once again we welcome you to a series of lecture notes in advanced calculus
course. This forms our first lecture unit in this course which is a continuation of lecture notes in
Calculus Courses we did in our first year. We would wish to mention that it is paramount that
you understand basic concepts of Calculus I, II and III before attempting this course. In this
lecture you are introduced to improper integrals and their applications covered in detail. Several
elaborate examples are given for illustration and sufficient exercises together with answers are
also given.



Objectives

At the end of this lecture you should be able to:
1 Define an improper integral
2 Solve problems concerning improper integrals with infinite limits of integration and
infinite discontinuities.
3 Apply improper integrals to real life situations and
4 State and apply the mean value theorem of integral calculus.


1.1. Definition and some examples of Improper Integrals

b
So far we have been dealing with an integral of the form  a f  x  dx where the interval [a,b] is
finite. In this section we consider an integral defined over an interval that is not finite in length
or one that has a finite number of discontinuities in the interval [a,b].

Definition 1.1.1

Integrals that satisfy either the properties

(a) One or both of the limits of integration are infinite, or
(b) f has a finite number of discontinuities in the interval [a,b] are called improper integrals.




1

, 1
Consider the area of the region under the graph of y  on the interval 1,  
x2
(see figure 1a).
y

1
y
x2




1a x
Fig 1a

This is of infinite extent which we have not considered before. Now consider the area in figure
1a above but say on a finite interval 1 to b (see figure 1b below).

y

1
y
x2




b x
1
Figure 1 b

b dx  1   1   1 
b

We evaluate area under the curve as  1 x 2
      
 x 1  b   1 
1
 1
b ____________ (1)

To determine area under the curve in figure 1a we let b approach infinity in equation 1. Thus
 dx b dx
1 x2 b1 x2
 lim

 1
 lim 1  
b  b

=1–0=1

This is an example of an infinitely long region with a finite area.




2

, 1
Now consider finding the area of the region under the graph of y  on the interval 1,   .
x
(See figure 2a)
y
y
1
y 1
x2 y
x2



1a x 1 b x

2a 2b
To evaluate area in figure 2a, first we evaluate area in 2b given by
k dx
1 x  ln x 1  ln k  ln1
k


 ln k _______________ (2)

and by use of equation 2 we have

 dx k dx
1 x k  1
 lim
x
 lim ln k  
k 

which increases indefinitely hence limit does not exist. This gives an example of an infinitely
long region with an infinite area.

Example 1.1.2

The integrals given by

 dx  dx
1 x
and 
0 x2
are improper because one of the limits of integration is infinite.

Example 1.1.3

The integrals given by

1 dx 3 dx
 0
1 x
and 3  x  1 2


are improper because the integrals have discontinuities somewhere in the interval of integration.



3

, For example the integral

1 dx
0 1 x
has a discontinuity at x =1 and hence not integrable there. Thus

1 dx m dx
0  lim 
1  x m1 0 1  x
m

 1
 lim 2 1  x  2 
m1  
0

 1 

 lim 2 1  m  2  2 
m1  

1
= 2 1  1 2  2  0  2  2 .

1.2. Improper Integrals with Infinite Limits of Integration

We define improper integrals with infinite limits of integration as follows.

Definition 1.2.1

 a,   ,
(a) If f is continuous on the interval  then
 b
a f  x  dx  blim
  a
f  x  dx


 , b  ,
(b) If f is continuous on the interval  then
b b
 f  x  dx  k 
lim  f  x  dx
k


 ,   ,
(c) If f is continuous on the interval  then

 c  c b
 f  x  dx   f  x  dx   f  x  dx  lim f  x  dx  lim
  f  x  dx
 c k  k b c
Where c is any real number.

In each case, if the limit exists, then the improper integral is said to converge; otherwise, the
improper integral diverges. In case (c) above, the improper integral on the left diverges if either
of the improper integrals on the right diverges.

We illustrate each of the three cases by use of the following examples



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