chapter 5 multiple integrals; applications of integration ( 다중적분 ; 적분의 응용 )...

Chapter 5 Multiple integrals; applications of integration ( 다중적분 ; 적분의 응용 )

Mathematical methods in the physical sciences 3rd edition Mary L. Boas

Lecture 16 Double & Triple integrals

- Use for integration : finding areas, volume, mass, moment of inertia, and so

on.

- Computers and integral tables are very useful in evaluating integrals.

1) To use these tools efficiently, we need to understand the notation and

meaning of integrals.

2) A computer gives you an answer for a definite integral.

1. Introduction

b

a

b

adxxfydx )(

AREA under the curve

AA

dxdyyxfdAyxf ),(),(

VOLUME under the surface “double integral”

2. Double and triple integrals ( 이중 , 삼중 적분 )

Example 1.

- Iterated integrals

AAA

dxdyydxdyzdAzV )1()(

AAA

dydxydxdyydxdyzV )1()1()(

2

22

0

222

0

22

0

264)2

()1( xxy

ydyyzdyxx

y

x

y

1

0

21

0

22

0 3

5)264(

xx

x

yA

dxxxdxzdyzdydx

2

0

2

0

2/1

0

2

0

2/1

0

3

5)2/1)(1(

)1()1(

y

y

y

y

y

xA

dyyy

dyyxdydxyzdxdy

(a)

(b)

‘Integration sequence does not matter.’

12 yx

Integrate with respect to y first,

A

b

ax

xy

xyy

dxdyyxfdxdyyxf)(

)(

2

1

),(),(

Integrate with respect to x first,

A

d

cy

yx

yxx

dydxyxfdxdyyxf)(

)(

2

1

),(),(

Integrate in either order,

dydxyxfdxdyyxfdxdyyxfd

cy

yx

yxxA

b

ax

xy

xyy

)(

)(

)(

)(

2

1

2

1

),(),(),(

In case of ),()(),( yhxgyxf

A

b

ax

d

c

b

a

d

cy

dyyhdxxgdydxyhxgdxdyyxf )()()()(),(

Example 2. mass=?

density

f(x,y)=xy

(0,0)

(2,1)

xydxdydxdyyxfdM ),(

11

0

2

0

2

0

1

0

yx xyA

ydyxdxxydxdydMM

Triple integral f(x,y,z) over a volume V, VV

dxdydzzyxfdVzyxf ),,(),,(

Example 3. Find V in ex. 1 by using a triple integral,

1

0

22

0

1

0

22

0

1

0

)1(x

x

yV x

x

y

y

z

dydxydydxdzdxdydz

Example 4. Find mass in ex. 1 if density =x+z,

dxdydzzxdM )(

2}1)23{(6/1}1)23{(2

2/)1()1(

)2

(

)(

1

0

32

1

0

22

0

2

1

0

22

0

1

0

2

1

0

22

0

1

0

dxxxx

dydxyyx

dydxz

xz

dydxdzzxdMM

x

x

x

y

x

x

y

y

z

x

x

y

y

zV

3. Application of integration; single and multiple integrals ( 적분의 응용 ; 단일적분 , 다중적분 )

Example 1. y=x^2 from x=0 to x=1

(a) area under the curve

(b) mass, if density is xy

(c) arc length

(d) centroid of the area

(e) centroid of the arc

(f) moments of the inertia

(a) area under the curve3

1

3

1

0

31

0

21

0

xdxxydxA

xx

(b) mass, if density of xy

1

0

5

0

1

0

1

00 12

1

2

22

x

x

yx x

x

yA

dxx

ydyxdxxydxdydMM

2xy

0 1

(c) arc length of the curve

dydydxdxdxdydydxds

dydxds

2222

222

)/(1)/(1

ds

dx

dy

(d) centroid of the area (or arc)

dxxdsxdx

dy 241 ,2

4

)52ln(5241

1

0

2 dxxdss

cf. centroid : constant

dA

xdAxxdAdAx ,

, , zdAdAzydAdAyxdAdAx

2xy

10

3

10

1

10or ,

4

3

4

1

4or ,

1

0

51

0 0

1

0 0

1

0

41

0 0

1

0 0

22

22

yx

AyydydxdydxydAy

xx

AxxdydxdydxxdAx

x

x

yx

x

y

x

x

yx

x

y

In our example,

mass of centroid: xdMdMx

arc of centroid : dsxdsx

(e)

If is constant,

1

0

221

0

21

0

2

1

0

21

0

2

414141

4141

dxxxdxxydxxydsy

dxxxdxxxdsx

(f) moments of the inertia

dxdydzrdMdMlI )(for ,2

dxdydzyxdMyxI

dxdydzxzdMxzI

dxdydzzydMzyI

z

y

x

)()(

)()(

)()(

2222

2222

2222

80

7)(

,16

1

2)(

,40

1

4)(

1

0 0

22

1

0

1

0

7

0

21

0 0

22

1

0

1

0

9

0

21

0 0

22

2

22

22

x

yx

x

y

z

x

x

yx

x

y

y

x

x

yx

x

y

x

IIxydydxyxI

dxx

xydydxxxydydxxzI

dxx

xydydxyxydydxzyI

In our example, (=xy)

yxz IIIcf .

EX. 2 Rotate the area of Ex. 1 (y=x^2) about x-axis

(a) volume

(b) moment of inertia about x axis

(c) area of curved surface

(d) centroid of the curved volume

(a) volume

5

1

0

41

0

2 dxxdxyV(i)

(ii)

22

2424 to

xzx

zxyzxy

dxdydzV

1

0

2

2

24

24x

x

xz

zxy

zxy

dydzdxV

(b) I_x (=const.)

MdydzdxzydVzyIx

xz

xz

zxy

zxy

x 18

5

18)()(

1

0

2222

2

2

24

24

(c) area of curved surface

ydsdA 2

1

0

221

0

4122xx

dxxxydsA

(d) centroid of surface

1

0

2x

ydsxxdAAx

Chapter 5 Multiple integrals: applications of integration

Mathematical methods in the physical sciences 3rd edition Mary L. Boas

Lecture 17 Change of variables in integrals

4. Change of variables in integrals: Jacobians ( 적분의 변수변환 ; Jacobian)

In many applied problems, it is more convenient to use other coordinate

systems instead of the rectangular coordinates we have been using.

sin

cos

ry

rx

- polar coordinate:

dxdydA1) Area

rdrdrddr

2) Curve 222 dydxds 22 )( rddr

drdr

drdr

d

drds 2222 )(1)(

Example 1 r=a, density

(a) centroid of the semicircular area 0 . ycf

Ad

Axdx

2

00

2/

2/ 2ardrrdrd

dxdydA

a

r

a

r

3

22cos))(cos(

3

0

2

0

2/

2/

2

0

2/

2/

adrrdrdrrdrdrxdA

a

r

a

r

a

r

3

4

2

2

2

32 ax

aaxxdAdAx

(b) moment of inertia about the y-axis

8cos

)(

4

0

222/

2/

222222

ardrdr

rdrdxdxdyxdxdydzxdMxdMzxI

a

r

y

,2

2

0

2/

2/

ardrdrdrdM

a

r

48

2 24

2

Maa

a

MI y

- Cylindrical coordinate

- Spherical coordinate

22222

sin

cos

dzdrdrds

dzrdrddV

zz

ry

rx

2222222

2

sin

sin

cos

sinsin

cossin

drdrdrds

ddrdrdV

rz

ry

rx

Jacobians (Using the partial differentiation)

t

y

s

yt

x

s

x

ts

yx

ts

yxJJ

),(

),(

,

,

dsdtJdAdxdy

rr

ry

r

y

x

r

x

r

yx

cossin

sincos

),(

),( rdrddxdy

t

w

s

w

r

wt

v

s

v

r

vt

u

s

u

r

u

tsr

wvuJ

),,(

),,( drdsdtJtsrfdudvdwwvuf ),,(),,(

** Prove that ddrdrdV sin2

y

x

z

r=h

h

Example 2. ? and ? zIz

322

3

0

2

0 0

2

0

hdz

zdzrdrddVM

hh

z

z

r

4

3

43

,42

2

43

4

0

2

0 0

2

0

hz

hhz

hdz

zz

dzzrdrdzdVdVz

h

h

z

z

r

25

0

4

0 0

22

0 10

3

1042 Mh

hdz

zdzrdrdrI

hh

z

z

r

z

Mass:

Centroid:

Moment of inertia:

Example 3. Moment of inertia of ‘solid sphere’ of radius a

3

44

3

sin

33

2

0 0

2

0

aa

ddrdrdVMa

r

15

82

3

4

5

sin)sin()(

55

2

0 0

222

0

22

aa

ddrdrrdMyxIa

r

2

5

2MaI z

2222222

2

sin

sin

cos

sinsin

cossin.

drdrdrds

ddrdrdV

rz

ry

rxcf

Example 4. I_z of the solid ellipsoid 12

2

2

2

2

2

c

z

b

y

a

x

1''' then ,',',' 222 zyxczzbyyaxx

',',' cdzdzbdydyadxdx

1) radius of sphere of volume(''' abcdzdydxabcM

abcabcM 3

41

3

4 3

In a similar way,

')''()( 222222 dVybxaabcdVyxI

22222222 '''' where ,''3

1'''''' zyxrdVrdVzdVydVx

5

4''4

)''''sin'('''

1

0

4

2

0 0

221

0

2

drr

dddrrrdVrr

5

4

3

1)('''' 222222 baabcdVybdVxaabcI

)(5

1 22 baMI

5. Surface integrals (?) ( 표면적분 )

dxdydAdAdxdy sec ,cos ‘projection of the surface to xy plane’

dxdydA sec kncos

surface tonormal),,( grad z

ky

jx

izyx

.),,( constzyx

gradgradn /)(

cos/

grad

z

grad

gradkkn

z

zyx

z

grad

kn

/

)()()(

/

1

cos

1sec

222

1 so

),,(),,( ),,(For

z

yxfzzyxyxfz

1)()(cos

1sec 22

y

f

x

f

Example 1. Upper surface of the sphere by the cylinder

0 ,1 22222 yyxzyx

.),,( constzyx

222),,( zyxzyx

22

222

1

11)2()2()2(

2

1

/sec

yxzzyx

zz

grad

1 to0 from

to0 from 2

y

yyx

1

022

0 12

2

y

yy

x yx

dxdy

/2 0 from

sin to0 from

r

2)cos1(2)1sin1(2

121

2

2/

0

2/

0

2

2/

0

22/

0

2/

02

sin

0

dd

drr

rdrd

x

H. W. (due 5/28)

Chapter 5

2-43 3-17, 18, 19, 20 4-4