Ch12 Multiple Integrals

8/10/2019 Ch12 Multiple Integrals

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ESSENTIAL

CALCULUS

CH12 Multipleintegrals


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In this Chapter:

12.1 Double Integrals over Rectangles

12.2 Double Integrals over General Regions

12.3 Double Integrals in Polar Coordinates

12.4 Applications of Double Integrals

12.5 Triple Integrals

12.6 Triple Integrals in Cylindrical Coordinates

12.7 Triple Integrals in Spherical Coordinates12.8 Change of Variables in Multiple Integrals

Review


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Chapter 12, 12.1, P665

(See Figure 2.) Our goal is to find

the volume of S.

}),(),,(0),,{( 3 RyxyxfzRzyxs


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Chapter 12, 12.1, P667


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Chapter 12, 12.1, P667


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Chapter 12, 12.1, P667

5. DEFINITION The double integral of f overthe rectangle R is

if this limit exists.

ij

m

i

n

jRyx

AjyijxifdAyxfii

),(lim),(

1

*

1

*

0,m ax


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Chapter 12, 12.1, P668

AyxfdAyxfm

i

n

j

ji

R

nm

),(lim),(

1 1

,


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Chapter 12, 12.1, P668

If f (x, y) 0, then the volume V of the solid thatlies above the rectangle R and below the surface

z=f (x, y) is

R

dAyxfv ),(


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Chapter 12, 12.1, P669

MIDPOINT RULE FOR DOUBLE INTEGRALS

wherexiis the midpoint of [xi-1,xi] and yjis the

midpoint of [yj-1, yj].

AyxfdAyxf ji

m

i

n

jR

),(),(1 1


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Chapter 12, 12.1, P672

10. FUBINIS THEOREM If f is continuous onthe rectangle R={(x, y} a x b, c y d} ,

then

More generally, this is true if we assume that fis bounded on R, is discontinuous only on afinite number of smooth curves, and theiterated integrals exist.

b

a

d

c

d

c

b

aR

dxdyyxfdydxyxfdAyxf ),(),(),(


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Chapter 12, 12.1, P673

R

b

a

d

c dyyhdxxgdAyhxg )()()()( where R ],[],[ dcbaR


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Chapter 12, 12.2, P676


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Chapter 12, 12.2, P676


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Chapter 12, 12.2, P677


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Chapter 12, 12.2, P677


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Chapter 12, 12.2, P677

3.If f is continuous on a type I region D such that

then

)}()(,),{( 21 xgyxgbxayxD

D

b

a

xg

xgdydxyxfdAyxf

2

1

),(),(


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Chapter 12, 12.2, P678


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Chapter 12, 12.2, P678

D

d

c

yh

yhdxdyyxfdAyxf

)(

)(

2

1

),(),(

where D is a type II region given by Equation 4.


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Chapter 12, 12.2, P681


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Chapter 12, 12.2, P681

D D D

dAyxgdAyxfdAyxgyxf ),(),()],(),([

D D

dAyxfcdAyxcf ),(),(

D g

dAyxgdAyxf ),(),(

6.

7.

If f (x, y) g (x, y) for all (x, y) in D, then8.

If D=D1U D2, where D1 and D2dont overlap

except perhaps on their boundaries (see Figure17), then

D D D

dAyxfdAyxfdAyxf

1 2

),(),(),(


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D

DAdA )(1

Chapter 12, 12.2, P682


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Chapter 12, 12.2, P682

11. If m f (x, y) M for all (x ,y) in D, then

D

DMAdAyxfDmA )(),()(


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Chapter 12, 12.3, P684


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Chapter 12, 12.3, P684


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Chapter 12, 12.3, P684

r2=x2+y2 x=r cos y=r sin


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Chapter 12, 12.3, P685


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Chapter 12, 12.3, P686

2. CHANGE TO POLAR COORDINATES IN ADOUBLE INTEGRAL If f is continuous on a

polar rectangle R given by 0arb, a,where 0-2 , then

R

b

a

rdrdrrfdAyxf

)sin,cos(),(


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Chapter 12, 12.3, P687

3. If f is continuous on a polar region of the form

then

)()(,),{( 21 hrhrD

Dh

h rdrdrrfdAyxf

)(

)(

2

1 )sin,cos(),(


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Chapter 12, 12.5, P696

3. DEFINITION The triple integral of f overthe box B is

if this limit exists.

B

l

i

m

j

ijkijkijk

n

k

ijkzyx

VzyxfdVzyxfkii 1 1

**

1

*

0,,m ax),,(lim),,(


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Chapter 12, 12.5, P696

B

l

i

m

j

kj

n

k

inml

VzyxfdVzyxf1 1 1

,,),,(lim),,(


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Chapter 12, 12.5, P696

4. FUBINIS THEOREM FOR TRIPLEINTEGRALS If f is continuous on therectangular box B=[a, b][c, d ][r, s], then

B

s

r

d

c

b

adxdydzzyxfdVzyxf ),,(),,(


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Chapter 12, 12.5, P697

dAdzzyxfdVzyxfE D

yxu

yxu

),(

),(

2

1

),,(),,(


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Chapter 12, 12.5, P697

b

a

xg

xg

yxu

yxuE

dzdydxzyxfdVzyxf)(

)(

),(

),(

2

1

2

1

),,(),,(


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Chapter 12, 12.5, P698

d

c

xh

xh

yxu

yxuE

dzdxdyzyxfdVzyxf)(

)(

),(

),(

2

1

2

1

),,(),,(


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Chapter 12, 12.5, P698

dAdxzyxfdVzyxfE D

yxu

yxu

),(

),(

2

1

),,(),,(


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Chapter 12, 12.5, P699

dAdyzyxfdVzyxfE D

yxu

yxu

),(

),(

2

1

),,(),,(


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Chapter 12, 12.5, P700

E dVEV )(


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Chapter 12, 12.6, P705


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Chapter 12, 12.6, P705

To convert from cylindrical to rectangularcoordinates, we use the equations

1 x=r cos y=r sin z=z

whereas to convert from rectangular to cylindrical

coordinates, we use

2. r2=x2+y2 tan = z=zx

y


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Chapter 12, 12.6, P706


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Chapter 12, 12.6, P706


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Chapter 12, 12.6, P706


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Chapter 12, 12.7, P709

0p 0


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Chapter 12, 12.7, P709


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Chapter 12, 12.7, P709


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Chapter 12, 12.7, P709


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Chapter 12, 12.7, P710


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Chapter 12, 12.7, P710

cossinpx sinsinpy cospz


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Chapter 12, 12.7, P711

where E is a spherical wedge given by

E

dVzyxf ),,(

d

c

b

addpdppsomppf

sin)cos,sin,cossin( 2

},,),,{( dcbpapE

Formula for triple integration in sphericalcoordinates


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Chapter 12, 12.8, P716


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Chapter 12, 12.8, P717


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9 CHANGE OF VARIABLES IN A DOUBLE


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Chapter 12, 12.8, P719

9. CHANGE OF VARIABLES IN A DOUBLEINTEGRAL Suppose that T is a C1

transformation whose Jacobian is nonzero and

that maps a region S in the uv-plane onto aregion R in the xy-plane. Suppose that f iscontinuous on R and that R and S are type I ortype II plane regions. Suppose also that T is

one-to-one, except perhaps on the boundary of .S. Then

R S dudvvu

yx

vuyvuxfdAyxf ),(

),(

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


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Ch12 Multiple Integrals