Multiple Integration

8/7/2019 Multiple Integration

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Chapter Multiple Integration

Dr. Tran Van Long

24-01-2011

Dr. Tran Van Long () Chapter Multiple Integration 24-01-2011 1 / 15
http://find/http://goback/


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Double Integrals

1 Double Integrals

2 Double Integrals in Cartesian Coordinates

3 Change of Variables in Double Integrals

4 Triple Integrals

5 Change of Variables in Triple IntegralsDr. Tran Van Long () Chapter Multiple Integration 24-01-2011 2 / 15


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Double Integrals

Double Integrals over Rectangles

D- a closed rectangle D= [a, b]

[c, d]



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Double Integrals


D- a closed rectangle D= [a, b] [c, d]function f(x, y) is bounded on D


D bl I l


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Double Integrals


D- a closed rectangle D= [a, b] [c, d]function f(x, y) is bounded on DThe partition P:

a = x0 < x1 < < xm = b,

c= y0 < y1 < < yn = d.


D bl I t l


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Double Integrals



a = x0 < x1 < < xm = b,

c= y0 < y1 < < yn = d.

Pconsists mn rectangles Rijand has area dxidyj= (xi xi1)(yj yj1)


Double Integrals


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Double Integrals



a = x0 < x1 < < xm = b,

c= y0 < y1 < < yn = d.


The norm of partition P: P = maxi,j

dx2i+ dy

2j


Double Integrals


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Double Integrals



a = x0 < x1 < < xm = b,

c= y0 < y1 < < yn = d.


The norm of partition P: P = maxi,j

dx2i+ dy

2j

I=D

f(x, y)dxdy=D

f(x, y)dA = limP0

i,j

f(xi, yj),

where (xi, yj) arbitrary point in Rij.


Double Integrals


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Double Integrals

Double Integrals over General Domains

f(x, y) is defined on bounded domain D, let f(x, y) is defined on rectangleR Dand be an extension offthat is zero outside D.


Double Integrals


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Double Integrals

Double Integrals over General Domains

f(x, y) is defined on bounded domain D, let f(x, y) is defined on rectangleR Dand be an extension offthat is zero outside D.

D

f(x, y)dxdy=

R

f(x, y)dxdy


Double Integrals


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g

Properties of Double Integral

Area area(D) =D

dxdy


Double Integrals


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g


Area area(D) =D

dxdy

Linear D

[af(x, y) + bg(x, y)]dxdy=

aD

f(x, y)dxdy+ bD

g(x, y)dxdy


Double Integrals


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g


Area area(D) =D

dxdy

Linear D

[af(x, y) + bg(x, y)]dxdy=

aD

f(x, y)dxdy+ bD

g(x, y)dxdy

Additivity of domains D1,D2 nonoverlapping

D1D2f(x, y)dxdy=

D1f(x, y)dxdy+

D2f(x, y)dxdy


Double Integrals in Cartesian Coordinates


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Fubinis Theorem

D= [a, b] [c, d] = {(x, y) : a x b, c y d}:

D

f(x, y)dxdy=b

a

dxdc

f(x, y)dy




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Fubinis Theorem

D= [a, b] [c, d] = {(x, y) : a x b, c y d}:

D

f(x, y)dxdy=b

a

dxdc

f(x, y)dy

D= {(x, y) : a x b, c(x) y d(x)}:

D

f(x, y)dxdy=

ba

dx

d(x)c(x)

f(x, y)dy




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Fubinis Theorem

D= [a, b] [c, d] = {(x, y) : a x b, c y d}:

D

f(x, y)dxdy=b

a

dxdc

f(x, y)dy

D= {(x, y) : a x b, c(x) y d(x)}:

D

f(x, y)dxdy=

ba

dx

d(x)c(x)

f(x, y)dy

D= {(x, y) : c y d, a(y) x b(y)}:

D

f(x, y)dxdy=

dc

dy

b(y)a(y)

f(x, y)dx




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Example 1

Change the order of the integral

D

f(x, y)dxdy,

Dis the triangle with vertices(

0, 0),

(1, 0

),

(1, 1

)




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Example 1


D

f(x, y)dxdy,

Dis the triangle with vertices(

0, 0),

(1, 0

),

(1, 1

)D

f(x, y) =

10

dx

x0

f(x, y)dy,




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Example 1


D

f(x, y)dxdy,

Dis the triangle with vertices (0, 0), (1, 0), (1, 1)

D

f(x, y) =

10

dx

x0

f(x, y)dy,

D

f(x, y) =

10

dy

1y

f(x, y)dx.




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Example 2

Find D

(x y)dxdy,

Dis the region bounded by y= 2 x2, y= 2x 1




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Example 2

Find D

(x y)dxdy,


D

(x y)dxdy=

13

dx

2x22x1

(x y)dy




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Example 2

Find D

(x y)dxdy,


D

(x y)dxdy=

13

dx

2x22x1

(x y)dy

=

1

3

dxxy

1

2y22x2

2x1



E l 2


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Example 2

Find D

(x y)dxdy,


D

(x y)dxdy=

13

dx

2x22x1

(x y)dy

=

1

3

dxxy

1

2y22x2

2x1

=13

x(2 x2) 12 (2 x

2)2x(2x 1) 12 (2x 1)

2dx=



E l 2


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Example 2

Find

D

(x y)dxdy,


D

(x y)dxdy=

13

dx

2x22x1

(x y)dy

=

1

3

dxxy

1

2y22x2

2x1

=13

x(2 x2) 12 (2 x

2)2x(2x 1) 12 (2x 1)

2dx= 6415



E l 3


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Example 3

Find the volume of the solid bounded by the planes:

z= 0, z= 1 x2, y= 0, y= x



E l 3


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Example 3


z= 0, z= 1 x2, y= 0, y= x

V= 0x1,0yx

(1 x2)dxdy



E l 3


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Example 3


z= 0, z= 1 x2, y= 0, y= x

V= 0x1,0yx

(1 x2)dxdy

V=

1

0

dx

x

0

(1 x2)dy



Example 3


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Example 3


z= 0, z= 1 x2, y= 0, y= x

V= 0x1,0yx

(1 x2)dxdy

V=

1

0

dx

x

0

(1 x2)dy

V=1

4


Change of Variables in Double Integrals

Transformation


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Transformation

Mapping: F: (u, v) (x= x(u, v), y= y(u, v)) maps one point (u, v) inuv-plane to one and only one point (x, y) in xy-plane and vice versa. TheJacobian

(x, y)

(u, v)

=

det

xu

yu

xv

yv



Transformation


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Transformation

Mapping: F: (u, v) (x= x(u, v), y= y(u, v)) maps one point (u, v) inuv-plane to one and only one point (x, y) in xy-plane and vice versa. TheJacobian

(x, y)

(u, v)

=

det

xu

yu

xv

yv

The transformation F(S) = D, where Sdomain uv-plane, Ddomainxy-plane.

F(S)

f(x,y)dxdy=

S

fF(u

,v)(x, y)(u, v)

dudv



Polar Coordinates


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Polar Coordinates

Polar coordinates [r, ], (r 0, 0 2) and Cartesian coordinates(x, y) are related by the transformation

F: [r, ] (x= rcos , y= rsin )

.



Polar Coordinates


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Polar Coordinates

Polar coordinates [r, ], (r 0, 0 2) and Cartesian coordinates(x, y) are related by the transformation

F: [r, ] (x= rcos , y= rsin )

.

The Jacobian (x, y)

(u, v)

=

cos rsin sin rcos

= r



Example 1


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Example 1

Find the area of the disk D= {(x, y) : x2 + y2 R2}.



Example 1


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Example 1


Area(D) =

D

dxdy



Example 1


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Example 1


Area(D) =

D

dxdy

Transformation x= rcos , y= rsin , where 0 r R, 0 2

Area(D) =

D

dxdy=

2

0

d

R

0

rdr= R2.



Example 2


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Example 2

Find the area of the elipse E= {(x, y) : x2

a2+ y

2

b2 1}.



Example 2


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p


a2+ y

2

b2 1}.

Area(E) =

E

dxdy



Example 2


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p


a2+ y

2

b2 1}.

Area(E) =

E

dxdy

Transformation x= au, y= bv, where u2 + v2 1,(x,y)(u,v)

= abArea(E) =

E

dxdy=

u2

+v2

1

abdudv= ab.



Area of surface


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The area of surface z= f(x, y) defined for (x, y) D:

S=

D

1 + (zx)

2 + (zy)2dxdy

Example: Find area of surface of unit sphere.

z=

1 x2 y2, x2 + y2 1

S= 2D

1 + (zx)2 + (zy)2dxdy= 2

D

11 x2 y2dxdy


Triple Integrals


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Change of Variables in Triple Integrals


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Multiple Integration