Integral Susun

8/12/2019 Integral Susun

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Chapter 5 Multiple integrals; applications of integration

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


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- 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


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b

a

b

adxxfydx )(

AREA under the curve

AA dxdyyxfdAyxf ),(),(

VOLUME under the surface

double integral

2. Double and triple integrals


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Example 1.

- Iterated integrals

AAA

dxdyydxdyzdAzV )1()(


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AAA

dydxydxdyydxdyzV )1()1()(

2

22

0

222

0

22

0

264)2

()1( xxy

ydyyzdy

xx

y

x

y

1

0

2

1

0

22

035)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


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Integrate with respect toyfirst,

A

b

ax

xy

xyy

dxdyyxfdxdyyxf

)(

)(

2

1

),(),(

Integrate with respect toxfirst,

A

d

cy

yx

yxx

dydxyxfdxdyyxf

)(

)(

2

1

),(),(


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Integrate in either order,

dydxyxfdxdyyxfdxdyyxf

d

cy

yx

yxxA

b

ax

xy

xyy

)(

)(

)(

)(

2

1

2

1

),(),(),(


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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 ),(

1

1

0

2

0

2

0

1

0

yx xyA

ydyxdxxydxdydMM


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

dxdydzzyxfdVzyxf ),,(),,(

Example 3. Find Vin 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


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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

dxxx

x

dydxyyx

dydx

z

xz

dydxdzzxdMM

x

x

x

y

x

x

y

y

z

x

x

y

y

zV


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3. Application of integration; single and multiple integrals

Example 1.y=x^2fromx=0 tox=1

(a) area under the curve(b) mass, if density isxy

(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

2

1

0

xdxxydxA

xx

(b) mass, if density ofxy

1

0

5

0

1

0

1

0012

1

2

22

x

x

yx x

x

yA

dxx

ydyxdxxydxdydMM

2xy

0 1


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(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


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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,

massofcentroid: xdMdMx

arcofcentroid: dsxdsx

(e)

If is constant,

1

0

22

1

0

2

1

0

2

1

0

2

1

0

2

414141

4141

dxxxdxxydxxydsy

dxxxdxxxdsx


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(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

2

1

0 0

22

1

0

1

0

9

0

2

1

0 0

22

2

22

22

x

yx

x

y

z

x

x

yx

x

yy

x

x

yx

x

y

x

IIxydydxyxI

dx

x

xydydxxxydydxxzI

dxx

xydydxyxydydxzyI

In our example, (=xy)

yxz IIIcf .


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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

4

1

0

2 dxxdxyV(i)

(ii)

22

2424 to

xzx

zxyzxy

dxdydzV

1

0

2

2

24

24x

x

xz

zxy

zxy

dydzdxV


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(b) I_x (=const.)

MdydzdxzydVzyIx

xz

xz

zxy

zxy

x18

5

18)()(

1

0

2222

2

2

24

24

(c) area of curved surface

ydsdA 2

1

0

22

1

0

4122xx

dxxxydsA

(d) centroid of surface

1

0

2x

ydsxxdAAx


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Chapter 5 Multiple integrals: applications of integration

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


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4. Change of variables in integrals: Jacobians ( ; 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:

dxdydA 1) Area

rdrdrddr

2) Curve 222 dydxds

22 )( rddr

drdr

drdr

d

drds 2222 )(1)(


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Example 1 r=a, density

(a) centroid of the semicircular area 0. ycf

AdAxdx

2

00

2/

2/2 ardrrdrd

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


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(b) moment of inertia about the y-axis

8cos

)(

4

0

22

2/

2/

222222

ardrdr

rdrdxdxdyxdxdydzxdMxdMzxI

a

r

y

,2

2

0

2/

2/

ardrdrdrdM

a

r

48

2 24

2

Maa

a

MIy


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- Cylindrical coordinate

- Spherical coordinate

22222

sin

cos

dzdrdrds

dzrdrddVzz

ry

rx

2222222

2

sin

sin

cos

sinsin

cossin

drdrdrds

ddrdrdV

rz

ry

rx


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Jacobians (Using the partial differentiation)

t

y

s

yt

x

s

x

ts

yx

ts

yxJJ

),(

),(

,

,

dsdtJdAdxdy

rr

ry

r

y

x

r

x

ryx

cossin

sincos),(),( rdrddxdy

t

w

s

w

r

wtv

sv

rv

t

u

s

u

r

u

tsrwvuJ

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

** Prove that ddrdrdV sin2


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y

x

z

r=h

h

Example 2. ?and? zIz

322

3

0

2

0 0

2

0

hdzzdzrdrddVM

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

2

2

010

3

1042 Mh

hdz

zdzrdrdrI

hh

z

z

r

z

Mass:

Centroid:

Moment of inertia:


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

3

44

3

sin

33

2

0 0

2

0

aa

ddrdrdVM

a

r

15

82

3

4

5

sin)sin()(

55

2

0 0

222

0

22

aa

ddrdrrdMyxI

a

r

2

5

2MaIz

2222222

2

sin

sin

cos

sinsin

cossin.

drdrdrds

ddrdrdV

rz

ry

rxcf

222


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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)radiusofsphereofvolume(''' abcdzdydxabcM

abcabcM 3

41

3

4 3


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In a similar way,

')''()(222222 dVybxaabcdVyxI

22222222 ''''where,''31'''''' zyxrdVrdVzdVydVx

5

4''4

)''''sin'('''

1

0

4

2

0 0

22

1

0

2

drr

dddrrrdVrr

54

3

1)('''' 222222

baabcdVybdVxaabcI

)(5

1 22 baMI


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5. Surface integrals (?)

dxdydAdAdxdy sec,cos projection of the surface to xy plane

dxdydA sec kn

cos

surfacetonormal),,(gradz

k

y

j

x

izyx

.),,( constzyx

gradgradn /)(

cos

/

grad

z

grad

gradkkn

z

zyx

z

grad

kn

/

)()()(

/

1

cos

1sec

222


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1so

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

z

yxfzzyxyxfz

1)()(cos

1sec 22

y

f

x

f


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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

yxz

zyx

zz

grad

1to0from

to0from 2

y

yyx

1

022

0 12

2

y

yy

x yx

dxdy

/20from

sinto0from

r

2)cos1(2)1sin1(2

121

2

2/

0

2/

0

2

2/

0

2

2/

0

2/

02

sin

0

dd

drr

rdrd

x

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