Tr Tripl iple e In Integra tegrals ls Triple Integrals: ( - - PowerPoint PPT Presentation

Tr Tripl iple e In Integra tegrals ls

Triple Integrals:

( , , )

V

f x y z dv



Important properties of the Triple integral:

   

V V

a f x y z dv a f x y z dv 

 

, , , , ,

V V V

f x y z g x y z dv f x y z dv g x y z dv   

  

( ( , , ) ( , , )) ( , , ) ( , , ) .

1 2

V V V

f x y z dv f x y z dv f x y z dv  

  

( , , ) ( , , ) ( , , )

Evaluating the triple integral in rectangular coordinates

 

V

f x y z dv



, ,

   

       

2 2 1 1

x x y b V b x x y

f x y z dv f x y z dzdA

   



   

, ,

, , , ,

Example

Evaluate the iterated integral

 

V

x y dv 



Solution:

• ver the region V bounded by the planes:

0 , 0 , 0 , 1. x y z x y z      

 

1 1 1 x y x

x y dzdydx

  

 

  

 

1 1 1 z x y x z

x y z dydx

    

 

 

 

V

x y dv 



  

1 1

1

x

x y x y dydx



   

 

Remark:

If 1 f x y z  ( , , ) ,

V

V dv 

then the volume of the solid V is

If x y z ( , , ) , is density, then the mass of the solid V

V

m x y z dv   ( , , )

The coordinates of the center of the solid V are given by:

1

c V

x x x y z dv m    ( , , ) , 1

c V

y y x y z dv m    ( , , ) , 1

c V

z z x y z dv m  



( , , ) .

Example

Find the volume and the coordinates of the center of gravity of : the region bounded by the parabolic cylinder

Solution:

2

4 and 0 , 0 , 6 , 0. z x x y y z      

Assuming the density to be constant k.

 

2 2

6 6 2 4 2 4 y y x z x x z x z x y z x y

V dzdydx z dydx

             

 

    

  

6 2 2 6 2 2

4 4

y x x y y x y x

x dydx x y dx

       

   

  

 

2 2 2 3

1 6 4 6 4 32 3

x x

x dx x x

 

          



The mass of solid is:

 

V

m x y z dxdydz   , ,

32

V

k dxdydz kV k   



The coordinates of the centre of gravity:

V

k dxdydz 

1

c V

x xdzdydx m  



32

V

k xdzdydx k 



2

2 6 4 0 0

1 32

x

xdzdydx





  

4 3 

1

c V

y y dz dy dx m  



2

2 6 4 0 0

1 32

x

ydzdydx





  

32

V

k ydzdydx k 



3 

3 1 2

c V V

z k z dz dy dx z dz dy dx m k   

 

2

2 6 4 0 0

1 32

x

zdzdydx





  

8 5 