i) The document discusses change of order and change of variables for double integrals. ii) To change the order of integration, one sketches the region to determine the new limits. iii) A change of variables replaces x, y with functions of u and v, and the area element dx dy with the Jacobian du dv. iv) As an example, a change of variables is used to evaluate a double integral over a parabolic region in xy-plane in the uv-plane.

1. Topics :-
i) Change of order
ii) Change of variables

2.  Applicable when one integral is either difficult or impossible to
evaluate, whereas the other Integral can be evaluated easily.
 The change from one integral to the other is called change of
order of integration.
 For changing the order, we should sketch the region of
integration.
 From the sketch, the new limits of integration can be determined
as usual.
2

3. 3

4. ( , )
( , )
x x
x y x y x yu v
y yu v u v v u
u v
 
     
  
     
 
The Jacobian of the transformation T
given by x = g(u, v) and y = h(u, v) is:
( , )
( , )
x y
A u v
u v

   

4

5.  Replace x , y by their equivalent relations in
terms of u & v, the area element (dx dy) by
(J du dv) and the region R of integration in
xy-plane by region in the uv-plane.
5

6.  Use the change of variables x = u2 – v2,
y = 2uv to evaluate the integral
where R is the region bounded
by x-axis, parabolas y2 = 4 – 4x and y2 = 4
+ 4x, y ≥ 0.
R
y dA
6

7. 2 2
2 2( , )
2 2( , )
4 4 0
x x
u vx y u v
y y v uu v
u v
u v
 
  
 
 
 
  
1 1
2 2
0 0
1 1
3 3
0 0
1 14 2 31 1
4 2 00
1 13 2 4
00
( , )
2
( , )
(2 )4( )
8 ( )
8
(2 4 ) 2
R S
u
u
x y
y dA uv dA
u v
uv u v du dv
u v uv du dv
u v u v dv
v v dv v v





 
 
   
      
 
 
 

 7

8.  By:-
Group 3
1. 13BEMEG104
2. 13BEMEG105
3. 13BEMEG106
4. 13BEMEG107
5. 13BEMEG108
8