Multiple Integration

Unit no. 5 Multiple integration




INTRODUCTION TO DEFINITE INTEGRALS AND DOUBLE INTEGRALS

Definite Integrals:

The concept of definite integral

b
 f ( x)dx .....(1)
a
is physically the area under a curve y  f (x) ,(say), the
X-axis and the two ordinates x = a and x = b. It is defined
as the limit of the sum
f ( x1 )x1  f ( x2 )x2  f ( x3 )x3  f ( x4 )x4  .....  f ( xn )xn
When n   and each of the lengths x1,x2,x3, .......xn tends to zero.
Here x1,x2,x3, .......xn are n subdivisions into which the range of integration has been
divided and x1, x2, x3, .......xn are the values of x lying respectively in the 1st, 2nd, …, nth
subintervals.

Double Integrals
A double integral is the counter part of the above
definition in two dimensions.
Let f(x, y) be a single valued and bounded function of
two independent variables x and y defined in a closed
region A in xy plane. Let A be divided into n elementary

areas A1,A2,A3, .......An .
Let ( xr , yr ) be any point inside the rth elementary area
Ar .
Consider the sum

n
f ( x1 , y1 )A1  f ( x2 , y 2 )A2  .....  f ( xn , y n )An   f ( xr , y r )Ar .....(2)
r 1
Then the limit of the sum (2), if exists, as n and each sub-elementary area approaches

,to zero, is termed as ‘double integral’ of f(x, y) over the region A and expressed as
 f ( x, y )dA
A 


Thus,

n
 f ( x, y)dA  n lim

 f ( xr , yr )Ar
A Ar  0
r 1 


EVALUATION OF DOUBLE INTEGRAL IN CARTESIAN COORDINATES

Evaluation of double integral  f ( x, y )dx dy
R 
is discussed under following three possible cases:
Case I: When the region R is bounded by two continuous
curves y   (x) and y   (x) and the two lines (ordinates)
x = a and x = b.
In such a case, integration is first performed with
respect to y keeping x as a constant and then the
resulting integral is integrated within the limits x = a
and x = b.
Mathematically expressed as:

b   ( x) 
 f ( x, y )dy dx
 f ( x , y ) dx dy    
R a  ( x )  
Geometrically the process is shown in Fig. 5.3,
where integration is carried out from inner rectangle
(i.e., along the one edge of the ‘vertical strip PQ’ from
P to Q) to the outer rectangle.
Case 2: When the region R is bounded by two continuous
curves x   ( y ) and x   ( y ) and the two lines (abscissa)
y = a and y = b.
In such a case, integration is first performed with
respect to x. keeping y as a constant and then the
resulting integral is integrated between the two limits
y = a and y = b.

,Mathematically expressed as:

b   ( y)

 f ( x, y )dx dy     f ( x, y )dx dy

a  ( y )

R 
Case 3: When both pairs of limits are constants, the region
of integration is the rectangle ABCD (say).

In this case, it is immaterial whether f(x, y) is integrated first with respect to x or y, the
result is unaltered in both the cases.

1 1 x 2
1
Ex. 1 Evaluate   1  x  y22
dy dx
0 0 
Solution:
Let,
1 1 x 2
1
I=   dy dx
0 0 1  x2  y2
Clearly, here y = f(x) varies from 0 to 1  x
2

and finally x (as an independent variable) goes between 0 to 1

1 1 x 2
1
I=  (  dy ) dx
0 0 (1  x 2 )  y 2 
Take 1  x  a
2 2


1 1 x 2
1
I=  (  dy ) dx
0 0 a2  y2 
1 x 2
1
1
1  y  
=   tan    dx
0 a  a  0 

   
 tan 1  1  x
1 2
=
1   tan 1 (0)  dx
0 1  x 2   1  x2




 

,  1
1
I=  dx
4 0 1 x 2


 {log( x  1  x 2 )}10
4
 



4
log 1  1  12  log 1  
 

I log(1  2 )
4

11
dx dy
Ex.2: Evaluate 
00 (1  x 2 )(1  y 2 ) 

Solution:

Let,

11
dx dy
I  
00 (1  x 2 )(1  y 2 ) 

The given integration is in the form

bb
  f ( x) g ( y ) dx dy
aa 

which can written as

b b
 f ( x)dx  g ( y )dy
a a

1 1
dx dy
I  
 0 (1  x 2 ) 0 1  y 2 