15.5

Triple Integrals in

Rectangular coordinates

, ,D D

FdV F x y z dxdydz Volume element: dV dzdydx

Rectangular Coordinates , , :x y z

Triple integrals where is a region is 3-space,

and the volume element

D

FdV D

dV

Volume of : ( ) 1D

D vol D dV

Review:

the region is typically described by:

( ) ( ),

R

g r h

( )

( )

area

h

g

rdrd

( , ) ( cos , sin )

R R

f x y dA f r r rdrd

Double Integrals in polar coordinates:

2 2 2 2 Find the volume bounded by + and 8 .y x z y x z Example :

VolumeD

dV 2 2

2 2

8 x z

R x z

dydA

2 28 2 2R

x z dA

use polar coordinates for the plane:x z

projection into the plane is a circle of radius 2x z

2 2

2

0 0

8 2Vol r rdrd

22

2 4

00

24 2 16 8 16

4r r d

2 2 2 2 2 2 the surfaces meet when + 8 or + 4y x z x z x z

15.6

Moments and center of mass

mass density = mass/area or mass/volume

mass density is a function of position: ( , , ) x y z

total mass: ( ) or ( )R D

M R dA M D dV

Mass:

Moment and center of mass:

moment = mass x distance

at the center of mass, moments must add to 0

mass are distributed on the axis at coordinate i im x x

if x is the center of mass, then need ( ) 0i im x x

or i i im x m x 1

or i i

i i

i

m xx m x

m M

total massiM m

if ( ) is mass density, and the center of mass, needx x

( ) 0x x dx or xdx xdx x dx x M xdx

xM

if ( , ) is mass density of a region in the x-y plane, total mass is R

x y R M dA

is moment around the axis,yM y

since distance to the axis is , we have: = y

R

y x M xdA

is moment around the axis:xM x

= x

R

M ydA

if ( , ) is the center of mass, then both

the moment around the axis and moment

around the axis must cancel:

x y

x

y

and y y xR R

R R

xdA ydAM M

xM MdA dA

,R

mass x y dA 1

0 0

x

xdydx 1

0 0

x

xy dx 1

3/2

0

x dx

15/2

0

2

5

x

2

5

,x

R

M y x y dA 1

0 0

x

xy dydx 1 2

0 02

x

xydx

1 2

0

2

xdx

13

06

x

1

6

,y

R

M x x y dA

1

2

0 0

x

x dydx 2

7

,

,y x

Center of Mass x y

M M

mass mass

5 5,

7 12

1

2

0 0

x

x y dx 1

5/2

0

x dx

17/2

0

2

7

x

(0.71,0.42)

Find the mass and center of mass of the lamina that

ocuppies the region bounded by , 0, 1 and has

mass densiy ( , )

y x y x

x y x

Example :

If mass density is constant, R R R

R R R

xdA xdA xdA

xdA dA dA

centroid: , ( ) ( )

R R

xdA ydA

x yarea D area D

if ( , , ) is mass density of a region in the 3-space, total mass is D

x y z D M dV is moment around the x-y plane

distance to the x-y plane is , hence:

xyM

z= xy

D

M zdV

= , == xz yz

D D

M ydV M xdV

, y yz xyxzD D D

D D D

xdV ydV zdVM MM

x zM M MdV zdV dV

center of mass ( , , ) :x y z

Centroid

mass density is a function of position: (x,y) or ( , , ) x y z

total mass: ( ) or ( )R D

M R dA M D dV

Summary: Center of mass and centroid

center of mass: , y D D D

D D D

xdV ydV zdV

x zdV zdV dV

1 1centroid (constant ): ,

( ) ( )R R

x xdA y ydAarea R area R

1 1 1centroid in 3-space: , ,

vol( ) vol( ) vol( )D D D

x xdV y ydV z zdVD D D

Average value of a function:

1 2

1The average value of quantities , ,..., is n ix x x x

n

The average value of a function ( ) over the interval [ , ] is

1( )

b

a

y f x a b

f x dxb a

The average value of a function ( , ) over a region R in the x-y plane

1is ( , )

( )R

z f x y

f x y dAarea D

The average value of a function ( , , ) over a region D in the 3-space

1is ( , , )

vol( )D

f x y z

f x y z dVD

15.7

Triple Integrals in

Cylindrical and spherical coordinates

, , , ,D D

F r z dV F r z rdzdrd

rdV dz drd

drdzdr

Cylindrical coordinates:

( , , ) ( , , )

cos( ), sin( )

x y z r z

x r y r

Volume element:

3 2

2 2

Evaluate where is the solid in the first octant

that lies beneath the paraboloid 1 .

E

x xy dV E

z x y

Example :

R

ESince the region in the plane is circular, we use cylindrical coordinates:xy2 2 20 1 0 1z x y rz

2 2 10 1z x y r

3 2 3 3 3 2cos cos sinx xy r r

3 2 2cos cos sinr 3 cosr

21 123

0 0 0

cos

r

r dzdr rd

02

2

1214

0

0 0

cosr

r z drd

12

4 2

0 0

1 cosr r drd

1 2

4 6

0 0

cosr r dr d

2

15 7

0

0

sin5 7

r r

1 1

15 7

7 5

35

2

35

Spherical coordinates:

distance to the origin

angle with the axisz

angle of the projection into

the x-y plane with the axisx

2 2 2 2certainly x y z

how do we convert this into x,y,z coordinates?

Spherical

Coordinates

, ,

sin cos

r

x

sin siny

cosz

2 2 2 2x y z

sinr r

r

2 2 2r z

0

, ,

z

r

z

sinr

cosz

0 2

Volume element :

2 sindV d d d

Volume of little box is =

width x depth x height

( ) ( )r

now use sinr

2

( ) ( sin( ) )

sin( )

Spherical

Coordinates

, ,

sin( )cos( )

sin( )sin( )

z cos( )

x

y

0

0 2

0

2 sindV d d d

2, , , , sinD D

F dV F d d d

: Compute the volume of a sphere of radius r.Example

2Vol = 1 sinD D

dV d d d 2

2

0 0 0

sin

r

d d d

3 2 3 3

000

1 1 4cos( ) 2 2

3 3 3

r

r r

/2

3

0

28sin( ) cos ( )sin( )

3d

Compute the volume between =cos( ) and

the hemisphere 2, 0.z

Example :

what is =cos( ) ? 2 cos( ) z 2 2 2or x y z z

2

2 2 1 1or

2 4x y z

1 1

a sphere of radius centered at 0,0, 2 2

thus cos( ) 2 and 0 , 0 22

21 sinD D

dV d d d 2 /2 2

2

0 0 cos( )

sin d d d

22 /2

3

cos( )0 0

1d sin( )

3d

/2

3

0

28 cos ( ) sin( )

3d

/2

4

0

2 18cos( ) cos ( )

3 4

2 18

3 4

31

6

3 31 4 4 1 16

easier: (2) ( )2 3 3 2 3 6

2

sin( )cos( )

sin( )sin( )

z cos( )

sin

x

y

dV d d d