Advanced Calculus and Complex Analysis

18MAB102T Advanced Calculus and Complex Analysis Multiple Integrals

Module – 1 Multiple Integrals
Evaluation of double integration Cartesian and plane polar coordinates – Evaluation of double integration
by changing order of integration – Area as a double integral (Cartesian) – Area as a double integral (Polar)
– Triple integration in Cartesian coordinates – Conversion from Cartesian to polar in double integrals –
Volume using triple integral – Application of Multiple integral in Engineering.

Evaluation of double integration – Cartesian and Polar coordinates


Type – 1 Limits are constants

  x 
1 2
1. Evaluate 2
 y 2 dx dy .
01

Solution:
2

 
12
 x3 
1
x  y dx dy     x y 2  dy
2 2

0 1
01
3

 8 
1
 1
    2 y 2     y 2  dy
0   3 
3
1
7 
    y 2  dy
0 
3
1
7 y3  8
  y   
3 3 0 3

  x 
21
8
Note: 2
 y 2 dy dx 
10
3

If the limits of integration are constants, then the order of integration is insignificant.
32
2. Evaluate   x y x  y  dy dx .
00

Solution:

 
32 32

  x y x  y  dy dx    x y  x y dy dx
2 2

00 00
2
 x2 y2
3
y3 
 
  x  dx
0
2 3 0
3
 8 
   2 x 2  x  dx
0
3 
3
 x3 8 x 2 
  2    30

 3 3 2 0



SRM IST, Ramapuram. 1 Department of Mathematics

, 18MAB102T Advanced Calculus and Complex Analysis Multiple Integrals


ab
dx dy
3. Evaluate  x y
.
22

Solution:
ab ab  dy
dx dy dx
  x y     x 
 y
22 22 
a
  log x b2
dy
2
y
 log x b2 log x a2
b a
 log   log  
2 2
32
4. Evaluate   r dr d .
00

Solution:

2
32
 r2 
3 3

  r dr d    2  d   2 d  2  0  6
  3

00 0 0 0




Type – 2 Limits are variables
1 x
5. Evaluate   x y x  y  dy dx .
0 x
Solution:

 x 
1 x 1 x

  x y x  y  dy dx   y  x y 2 dy dx
2

0 x 0 x
x
 x2 y2
1
y3 
 
  x  dx
0
2 3 x
1
 x3 x5/ 2 x 4 x 4 
       dx

0 
2 3 2 3
1
 
 x 4 x x5 x5  3
     
 8 3  7 10 15  56
 
 2 0




SRM IST, Ramapuram. 2 Department of Mathematics

, 18MAB102T Advanced Calculus and Complex Analysis Multiple Integrals


a a2  x2
6. Evaluate  y dy dx .
0 0

Solution:
a2  x2 a2  x2
a
 y2 a

  y dy dx    2 

0
dx
0 0 0
a
 a2  x2  a3
    dx 

0 
2 3


a ay
7. Evaluate   x y dx dy .
0 0

Solution:
a2  x2 ay
a
 x2 
a

  y dy dx    2 
y dy
0 0 0 0
a
1 a4
2 0
 y a y dy 
6


CHANGE THE ORDER OF INTEGRATION
For changing the order of integration in a given double integral
Step 1: Draw the region of integration by using the given limits.
Step 2: After changing the order, consider
 dxdy as horizontal strip
 dydx as vertical strip
Step 3: Find the new limits.
Step 4: Evaluate the double integral.
a a
xdydx
Change the order of integration in   2 and hence evaluate it.
8. x + y 2
0 y

Solution:
aa
x
  x 2  y 2 dx dy (Correct Form)
0 y

Given limits x : y  a ; y :0  a
After changing the order,
dy dx  vertical strip
Now, limit x :0  a ; y :0  x

a x
x
a x
 1 
  2 2
dydx    x  2  dy dx
0 0
x +y 0 0 
x + y2 


SRM IST, Ramapuram. 3 Department of Mathematics