Mu Multip ltiple le Int Integ egrals rals Double Integrals: - - PowerPoint PPT Presentation

Mu Multip ltiple le Int Integ egrals rals

Double Integrals:

( , ) ( , )

y d b d x b a c x a y c

f x y dydx f x y dy dx

   

        

  

Important properties of the double integral:

   

R

a f x y dA a f x y dA 

 

, ,        

R R R

f x y g x y dA f x y dA g x y dA       

  

, , , ,

     

1 2

R R R

f x y dx dy f x y dx dy f x y dx dy  

  

, , ,

Example

Evaluate the iterated integral:

2 3 2 1 0

x ydxdy



Solution:

3 2 2 3 1 1

9 3 x y dy ydy        

 

2 2 1

27 9 2 2 y        

2 2 3 3 2 2 1 0 1 y y

x ydxdy x ydx dy

 

      

  

Evaluating Double Integrals over General Regions

 

   

2 1

g x b D a g x

f x y dA f x y dy dx           

  

( , ) ,

 

   

2 1

h y d D c h y

f x y dA f x y dx dy         

  

( , ) ,

Example

Evaluate

Solution:

( 2 )

D x

y dA 



2 2

: 2 1 . D y x and y x   

2 2

1 1 1 2

( ( 2 2 ) )

x x y x D

x y dy x y dA dx

  

  

  

2 2

1 1 2 1 2 x x

xy y dx

 

     



1 2 3 4 1

(1 2 3 ) x x x x dx



    



32 15 

Example

Evaluate

Solution:

2

2 1 /2 y x

e dy dx



 

2 2

2 1 2 2 1 y y x y

e e dy dx dx dy

 



   

/

: /2, 1, 2, 0. D y x y x x    

2

1 2 x y y x

x e dy

  

      



2

1

2

y

y e dy



      



2 1

1

1 1 1

y

e e e

 

            

x

y

1 y 

2 y x / 

2 x 

The Double Integral in Polar Coordinate:

sin y r  

2 2

x y r  

cos x r  

dA rdrd 

Example

Evaluate

Solution:

2

(3 4 )

D

x y dA 



2 2 2 2

: 1 4. D x y and x y    

in the upper half-plane bounded by the circles

2 2 2 2 1

(3 4 ) (3 cos 4 sin )

R r

x y dA r r rdrd

 

  

 

  

  

2 2 3 2 1

(3 cos 4 sin )

r

r r drd

 

  

 

 

 

2 3 4 2 1

( cos sin ) r r d



    



2

15 15 (7cos 15sin ) (7cos (1 cos2 )) 2 2 d d

 

            

 

Applications of Double Integrals:

(1) Calculating the area of a plane region:

D

A dA 

(2) Calculating the Volumes:

( , ) .

D

V f x y dA  

Example

Calculate the area of a region bounded the curves:

Solution:

2

2 , . y x y x   

D

A dA 

 

2 2

1 2 1 2 2 2 x x x x

dy dx y dx

   

 

  

 

1 1 3 2 2 2 2

2 2 3 2 x x x x dx x

 

             



1 1 8 4 27 2 4 3 2 3 2 6       

Example

Calculate the volume of a solid bounded by the surfaces:

Solution:

0, 0, 1, 0. x y x y z z      

( , )

D

V f x y dA  

 

1

D

x y dA   



 

1 1

1

x

x y dydx



  

 

      

1 2 1 1 2

1 1 1 1 2 2

x

x y x y dx x x dx



               

 

   

1 2 3 1

1 1 1 2 6 6 x x dx              



Example

Find the volume of the solid bounded by the paraboloid:

Solution:

2 2

1 , 0. z x y z    

( , )

D

V f x y dA  

2 2

(1 )

D

x y dA   



2 1 2 0 0

(1 ) r rdrd



  

 

2 1 3

( ) d r r dr



  

 

1 2 4

2 2 4 2 r r           