Esercizi su integrali doppi: cambiamento di variabiliEs. 1. I = ZZ T e 2x y2 dxdy ove T = (x,y) 2...

Esercizi su integrali doppi: cambiamento di

variabili

Riccarda Rossi

Universita di Brescia

Analisi II

Riccarda Rossi (Universita di Brescia) Integrali doppi – Cambiamento di variabili Analisi II 1 / 105

Es. 1.

I =

ZZ

T

e

�x

2�y

2

dxdy

ove

T =

�

(x , y) 2 R2

: x

2

+ y

2 4, y � 0


^ 'tTfnegmmetaov

r.is

°

Cincolere

to> . feontiene

2 × Rterumime x4y2I

passoalleaeohd . pdoria e)


Jn coed . fdoriT → F

=G-

' a)= { ( r

,8) E to ,

to )xCoRI]|oerez

, Deion ] }= to ,z]x Toit ]

dxdy → 2dedO

×4Y2→r2 = Fao ) ipwdotodi una fz .

della

wheredrone

£= $eaz]×eo,

.se#drdo=TfzI


= ffosready . 1 bison]= I ( f2 - zr et ?

tz) de ]= it f£

.

) ( i" ] ? =

= ega . e- 4)

Es. 2.

I =

ZZ

T

y

2

dxdy

con

T =

�

(x , y) 2 R2

: 1 x

2

+ y

2 4, y � 0, x + y � 0


yz - x

m

^

YThe srmmetcia

. P cincohereas aeons

, Poitou#2>¥=G-'

it ).=Gr]×ete< 2 [934 'D0£ @ < 34T


it since ) rdrddI= §n⇒×ea3IiD

inner Cindy ( fakedSu Um

rutemsdo ;

Guyanese = ¥ B. facey#

a separate "

0


= real Eye]= if ( 3%5+12)

Es. 3.

I =

ZZ

T

x dxdy

ove

T =

�

(x , y) 2 R2

: y � 1, 1 (x � 2)

2

+ (y � 1)

2 4


÷he uncolore cona Y C = ( 2 , i ) con

roggi r e 22

1- 4

:>

2 ×


In word. polori T non si

save bone paehe A word.

Poutre sonoaolatte esrmmetaieuncdari Etmntoein C 0,07 .

Passage a delle word. polari

atraslateu cbeiflettamo besimmeieia cinwhere CEMRARA

nm ( 2,1 ).


Se

evessrmo1EX2tY2E4d1es2ws2otr2srm2OE4Quirecx-zj2tCy-n2e4.aews@1ksrnOLecoonD.P

OLARI " then ATE ' ' Soho legatea × e a y do :


X - 2 = Rusi # x= 2 trash )

y - i. rsrmO⇐D ytitksrmio )

⇒ Gen ,ot=( Give ,o )

, Gza ,o ))

con an c s,

ok reosiokrd

Gzch , e) = tesrmc 01+1

¥1IT G- a. a) I= kdrdi


NY MZRPRETAZ one :

4€;>>

k' " → ( XD

X X=n -2

Y=y - i

( 0,0 ) → ( 2cL )n

" "

y

e foipamo die ( X. Y ) alee usuali

Coors . POCARI


ffyx okay = ffpktrwsio))rdrdo

F : t.ro2,0 -0 EE

=$a. ]×ff⇒rtEytan↳ so

=3I ( eseriziu ! )

Es. 4.Siano T il triangolo di vertici A = (�2, 4), B = (2, 0), e C = (�2,�4) e

D =

�

(x , y) 2 T : x

2

+ y

2 � 1

Calcolare

I =

ZZ

D

x

2

dxdy


A =2§+x2dxdy• ay Ddsrmm

. tsp .

°

arse ×D\ . f e- PARI

.

'. i ;

93 >× bsp . ey2 fcxiyhfrxiy )

• @Dtednfyzo }


o

-^y4f¥¥Dt=It' C

+

¥14, ¥ He,

Tt : tuangolo converting ' mi

6,01,

C- 2,07,

C- 2,4 )Ct : semidrsw

: ×4y?cL, yzo


' §=,

iedxdy.

!qy74=2 - x

; ; K

Tt : nwunale hopeeto apex

Tt={ ( x. y ) / -2£×e2,

0 eye 2- × }§+×2dxdy=[z2$n×x@) dxdy =


= µ x2 ( 2 - x ) dx

ifz22x2dx-f.z2x3@0_4fo2x2dx-4fE3Po.s

}


1¥,×n Kid :

a →

E : or-4

0£ O E I

Kgtidxoy =D, ,

,×po2,

.gs?co)roeeoo=fo1r3dy.fegF2co5i=fizItg


ftp.x?dxd32g - Dg

§§2 dxdy = 2§jtd×dy

as →a

Es. 5.

I =

ZZ

D

(x + y) dxdy

ove

D =

n

(x , y) 2 R2

: x � 0, y 0,x

2

4

+ y

2 1

o


Up' one

roccwnseddM x÷←yE1

^Y a- 2

b=s< Gerd

. plan

@, 7× femeualizalez.io#x=2rasOy=rsimiotDDD~rEtoiD,OEL33ii2aJ


dxody

(2rTIohrsf*%)2rIrdo[=

¥0,1]×f3zI ,

at ] Q=2

b=1

( 4r2wsc a) t Qk2srm( d) dzdo=

§aDxt3zIRI]-

- IIn 2


4r2 Cosio ) dkdoIt §, ,+eEi 'D

= 4$10rzdy.fqyos.to)db

= 4- fSma ]at at

3zit

= 4--3


ZI

Iz =2§"

Mar. f since ) do

3zI

= § f- as ' offs -2g

⇒ I= Iit Iz = 2

3

Es. 6.

I = 2

ZZ

T

xy e

2y

2

x

2

+y

2

dxdy ,

dove

T = {(x , y) 2 R2

: x � 0 , y � 0 , 1 x

2

+ y

2 4}


^ it

F : rear ]T¥ a- a ⇒


1- 2$rwsarsm.ge#snwµ .[1,2%0,172]

.k§dO→ §

. .ae#gEEose2smgonoI ( smug )

'


=

15g lets )

Es. 7.ZZ

C

f (x , y) dx dy

con

C = {(x , y) 2 R2

: x

2

+ y

2 1},

f (x , y) =

(

3 se y � x ,

x se y < x


a.tw#jFefa.yax*ksfa.yl-ffgfaay)


§yfxy ) d * dy = § 3 dxdy

=3 area (G) = 317a- 2

area delsanidrseo


[email protected]

day - D=^ Y 3

9*de :

1 ?×# EL

¥;5*01*94*4


I=3zE+r§

Es. 8.

I =

ZZ

T

(x � y)

2

1 + (x � y)

2

dx dy

con

T = {(x , y) 2 R2

: 0 x 2, 0 x � y 2}


fdpeude dox. y 42×-2

# {yex

⇐fetobpeudino-y=x-z°da €y )¢

.

Tohipeude"

• •emelie do ×

2 → paso a move

coordinate ( a ,v )-2 tali In :


× . U

{×⇒vJl Cambrian

.

di VAREABILE :

III.manGzcu,v)=X

. V=U . W

|LGhvY=1


$,

¥¥gpdxey"

$p e¥fz1dadN →

= to ,2]xCo,2]T : OEX EZ OEKYEZ

rn to

F={o<uez,

oev £2 }=to,2]xEaz]


= 2 ( 2 - avian ( a ) ]

Es. 9.

I =

ZZ

T

y dx dy

con

T = {(x , y) 2 R2

: x � 0, x

2

+ y

2 2, y � x

2}


Oay

tµy=x2. Mangonia

womb.

I POURI anehe

•

• '

• . > term partiteb rz × do um disco :

Tnonfnesmm. wvdau


Of X £

bit1

b e- llexisse del ptookintense zone free Rey 2=2 e{ y=xz ,

cow Rnnedoxzo

, yzo

as be s

44×2<4 e pay /×FyE2YfIxtFf


I= ff§×F±ydy)dxy=Fx2= ft 1¥]y=

,

d×

= .. . . = #

15

Esercizi su integrali doppi: cambiamento di variabiliEs. 1. I = ZZ T e 2x y2 dxdy ove T = (x,y) 2...

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