Tich Phan Kep Phan 2

7/23/2019 Tich Phan Kep Phan 2

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ĐỔI BIẾN TRONG TÍCH PHÂN KÉP



TỌA ĐỘ CỰC

M

y

r

ϕx

[0, 2 ] [ , ]ϕ π ϕ π π ∈ ∈ −ha y

cos , sin

x r y r ϕ= =

2 2 0r x y = + ≥



TÍCH PHÂN KÉP TRONG TỌA ĐỘ CỰC

D

ϕ α =

ϕ β = Dij

j ϕ

1 j ϕ −

( )* *,i j r ϕ

ϕ ∆

:

a r b

D α ϕ β

≤ ≤

≤ ≤



Tổng tích phân

* * * * *

,

( cos , sin )n i j i j i i j

S f r r r r = ∆ ∆∑ ϕ ϕ ϕ

0( , ) lim n

d D

f x y dxdy S →

=∫∫

0lim ( cos , sin )nd D

S f r r rdrd → = ∫∫ ϕ ϕ ϕ



Công thức đổi bin !"ng t#" đ$ c%c

( , ) ( cos , sin )

D D

dxdy drd r f x y f r r =∫∫∫ ∫∫ ϕ ϕ

cos , sin x r y ϕ= =



Một số !"ng c#ng $% &'(n ) t*#ng t+, ộ c-c

2 2 2 x y R

r R

+ =

⇔ =

cos , sin

x r y r ϕ

= =

2 2 2

0

0 2

r R

x y R

≤ ≤

≤

+

⇔ ≤

≤

ϕ π

R

R

&R D&R R

R



•

2 2 2 x y Rx + = 2 2 2 x y Rx + ≤

•

R .R

2 cosr R =

ϕ

0 2 cos

2 2

r R ≤ ≤

−

≤ ≤

ϕ

π π

ϕ



•

2 2 2 x y Ry + =

0 2 sin

0

r R ≤ ≤≤ ≤

ϕ

ϕ π

•R

.R2 sinr R = ϕ

2 2 2 x y Ry + ≤



1( )r r = ϕ

2 ( )r r = ϕ

D 1 2( ) ( )

:r r r

Dϕ ϕ

ϕ

≤ ≤ ≤ ≤

(0 2 ) π< − ≤

( cos , sin )

D

f r r rdrd ∫∫ ϕ ϕ ϕ

2

1

( )

( )

( cos , sin )

r

r

d f r r rdr = ∫ ∫ ϕ

ϕ

ϕ ϕ ϕ



/Í )0

2 2

D

I x y dxdy = +∫∫

cos , sin

x r y r ϕ= =

2 2

1:0

x y Dy

+ ≤≥

12 Tính3 $4'

'&'

( ) '

0 1

0:

r D

≤ ≤ ≤ ≤ ϕ π

1

2

0 0

.

D

I r rdrd d r dr = =∫∫ ∫ ∫ π

ϕ ϕ

0

1

3 3d = =∫

π

π

ϕ



5 6

7 5 6

8 7

( ) *( ) '

1 2

34 4

:

r

D

≤ ≤ ≤ ≤π π

ϕ

cos , sin x r y r ϕ= =

( )

D

I x y dxdy = −∫∫ 2 21 4:,

x y Dy x y x

≤ + ≤≥ ≥ −

.2 Tính3



( cos sin ).

D

r r rdrd = −∫∫ ϕ ϕ ϕ

3

242

1

4

(cos sin )d r dr = −∫ ∫

π

π

ϕ ϕ ϕ

( )

D

I x y dxdy = −∫∫

34

4

8 1(cos sin )

3 3d

= − − ÷ ∫

π

π

ϕ ϕ ϕ

72

3= −

1 2

3

4 4

:

r

D

≤ ≤

≤ ≤

π π

ϕ



0 2sin

: 3

4

r

D

ϕ

π

ϕ π

≤ ≤

≤ ≤

* 6 .s'nϕ

2sin

2

3 0

cos cos

D

I r rdrd d r rdr = =∫∫ ∫ ∫ π ϕ

π

ϕ ϕ ϕ ϕ

1

6= −

cos , sin x r y r ϕ= =

D

I xdxdy = ∫∫ 2 2 2

: x y y

Dy x

+ ≤

≤ −

92 Tính3 $4'



2 2 2 2

4 , 2 , , 0 x y x x y x y x y + = + = = =

:2 Tính ;'<n tích &'(n ) g'4' h=n >?'3

cos , sin x r y r ϕ= =* 6 :c#sϕ

* 6 .c#sϕ

5 6

7

2cos 4cosr ϕ ϕ ≤ ≤

04

π ϕ ≤ ≤

)



( ) 1

D

S D dxdy = ∫∫

D

rdrd = ∫∫ ϕ

4cos4

0 2 cos

d rdr = ∫ ∫

π

ϕ

ϕ

ϕ

3 3

4 2

= +

2cos 4cosr ϕ ϕ ≤ ≤

04

π ϕ ≤ ≤

)



D

I xydxdy = ∫∫ 2 2

:

3 0

x y x D

x y

+ ≤ −

≤ ≤

@2 Tính3 $4'

* 6 8 c#sϕ

3y x =

0 cosr ϕ ≤ ≤ −

40

3

π ϕ ≤ ≤



ĐB BDN TNG EFT

( , ) ( , ) x y D u v D′∈ ⇔ ∈

D

+

,

7 6 7IJ$J 56 5IJ$

CLng thc ổ' >'n

1

( , )

( , )

J D u v

D x y

=

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

D D

f x y dxdy f x u v y v v J dud u

′

=∫∫ ∫∫

( , )

( , )

u v

u v

x x D x y J

y y D u v

′ ′= =

′ ′



p ;ng ổ' >'n tổng IQt

cos , sin

x r y r ϕ= =cos sin

sin cos

r

r

x x r J r

y y r

ϕ

ϕ

ϕ ϕ

ϕ ϕ

′ ′ −= = =

′ ′

( , ) ( cos , sin )

D D

f x y dxdy f r r rdrd

′

=∫∫ ∫∫ ϕ ϕ ϕ

T+, ộ c-c3



2 2 2

( , ) ( , ).1

D u v R

f x y dxdy g u v dudv

+ ≤

=∫∫ ∫∫

cos , sinu r v r ϕ= =

>I

)3 7 ,. S 5 >. ≤ R.

)"' gốc t+, ộ ntâ& 7 6 I S ,J 5 6 $ S >

Đổ' t'p s,ng

t+, ộ c-c3

•

, 7

5

1 01

0 1

u v

u v

x x J

y y

′ ′

= = =′ ′

Hnh t*Un tâ& tV5 W3$



•

,

>

7

5

I

$

)3 7 ,. S 5 >. ≤ R.

7 6 , S *c#sϕJ 5 6 > S *s'nϕ

X 6 *

TY& tZt3

* ϕ

0:0 2

r R Dϕ π

≤ ≤′ ≤ ≤

( , ) ( cos , sin )D D

f x y dxdy f a r b d r r rd ′

= + +∫∫ ∫∫ ϕ ϕ ϕ



Đổ' >'n t*#ng [\\'pps[ 2 2

2 2: 1 x y

Da b

+ ≤

),

> 7 6 ,*c#sϕJ 5 6 >*s'nϕX 6 ,>*

( , ) ( cos , sin )

D D

f x y dxdy f ar br drd abr

′

=∫∫ ∫∫ ϕ ϕ ϕ

0:

0

1

2

r D

≤ ≤′ ≤ ≤ ϕ π

2 22

2 2

x y r

a b+ =



7 6 . S *c#sϕJ 5 6 81 S *s'nϕ

X 6 *

(2 cos )( 1 sin )

D

I r r rdrd

′

= + − +

∫∫ ϕ ϕ ϕ

0 3

: 0

r

Dϕ π

≤ ≤′ ≤ ≤

-

.

D

I xydxdy = ∫∫ 12 Tính3

hnh t*Un3 7 ..

S 5 S 1.

≤ ]

$4' ) \% n^, t*_n c`,



(2 cos )( 1 sin )

D

I r r rdrd

′

= + − +∫∫ ϕ ϕ ϕ

3

2

0 0

( 2 cos 2 sin sin cos )d r r r rdr= − − + +

∫ ∫

π

ϕ ϕ ϕ ϕ ϕ

9 18= − +



/í ;

2 1

3 cos 0, 2 sin 0

r

r r ϕ

≤

≥ ≥

2 2

, : 1; 0; 09 4D

x y I xydxdy D y x = + ≤ ≥ ≥

∫∫ .2 Tính3

9

.7 6 9*c#sϕJ 5 6 .*s'nϕ

X 6 9a.a* 6 b*

M'(n ) !c $'t \='3



12

0 0

3 cos .2 sin .6

D

xydxdy d r r rdr

=∫∫ ∫ ∫

π

ϕ ϕ ϕ

9

2

=

0 1:

0

2

r D

≤ ≤′ ≤ ≤

π

ϕ

0 1

cos 0,sin 0

r

ϕ ϕ

≤ ≤⇔ ≥ ≥

2 1

3 cos 0, 2 sin 0

r

r r ϕ

≤

≥ ≥



•

92 Tính ;'<n tích &'(n g'4' h=n >?'2

2

1, 0, , 03

x

ellipse y y y x x + = = = ≥

3 cos , sin x r y r ϕ= =

M'(n ) !c $'t \='3

2

2 1, 03

x y y x + ≤ ≤ ≤

0 1,

0 sin 3 cos

r

r r

≤ ≤⇔

≤ ≤ ϕ ϕ

3J r =



0 1,

sin0 tan 3cos

r ≤ ≤

⇔ ≤ = ≤ϕ

ϕ

ϕ

0 1,

0 sin 3 cos

r

r r

≤ ≤

≤ ≤ ϕ ϕ

0 1

03

r

π

ϕ

≤ ≤⇔

≤ ≤

13

0 0

( ) 3

D

S D dxdy d rdr = =∫∫ ∫ ∫

π

ϕ



Tính ố' 7ng c`, &'(n ) t*#ng tính tp dep

D D'

) ố' 7ng I, #5

)1 6 ) ∩

f7J52 7 ≥

i7J5 chjn th[# 731

( , ) 2 ( , )

D D

f x y dxdy f x y dxdy =∫∫ ∫∫ i7J5 \k th[# 73 ( , ) 0

D

f x y dxdy =∫∫

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Tich Phan Kep Phan 2