8/10/2019 8_Areas of Bounded Regions

1/6

@^O@] IC NI_EJOJ ODAIE]0

www.tkaegaat.ae

@\\FAM@ZAIE] IC AEZOD^@F]

3.0 @^O@ _EJO^ ]AH\FO M_^SO

0. Zko `ro` ic tko rodaie niuejoj ny ` murvo y 8 c(x), x 8`, x 8 n, `ej tko x - `xas.

M`so ATkoe tko murvo y 8 c(x) faos nivo tko x-`xas.Zko ro` niuejoj ny tko murvo

y 8 c(x), tko x-`xas `ej tko irjae`tos x 8 ` `ej x 8 n as davoe ny

n

`jxy

y 8 c(x)

@

N

M J` n

Q

Q

I[ [

M`soAATkoe tko murvo y 8 c(x) faos nofiw tko x-`xas.Zko `ro` niuejoj ny tko

murvo y 8 c(x), tko x-`xas,`ej tko irjae`os x 8 ej x 8 n as davoe ny -n

`jx)y(

M` n

Q

Q

I[ [

@ N

M

Jy8c(x)

>. Zko @ro` ic tko odaie Niuejoj ny Murvo x 8 c(y), Zko @nsmass`o y 8 m, y 8 j `ej

tko y -`xas.

M`so A Tkoe tko murvo x 8 c(y) faos ti tko radkt ic tko y -`xas.Zko ro` niuejoj ny

tko murvo x 8 c(y), tko y-`xas`ej tko `nsmass`o y 8 m `ej y 8 j as davoe ny j

m

jyx

y8j

y8m

J

M

x8c(y)N

@

Q

Q

[ [

M`so AA Tkoe tko murvo x 8 c(y) faos ti tko foct ic tko y-`xas.Zko ro` niuejoj ny

tko murvo x 8 c(y), tko y-`xas `ej tko `nsass` y 8 m `ej y 8 j as davoe ny -j

m

jy)x(

Q

y8j

y8m

x8c(y)

I[

Q

[

8/10/2019 8_Areas of Bounded Regions

2/6

@^O@] IC NI_EJOJ ODAIE]=

www.tkaegaat.ae

3.> ]IHO ]Z@EJ@^J M_^SO]

0. Marmfo

(a) x>+ y>8 `>roprosoets ` marmfo watk moetro `t (8 > roprosoets marmfo watk

N(cy + m 8 < roprosoets tko doeor`f oqu`taie ic ` marmfo watk moetro `t

(d, c) `ej r`jaus 8 mcd >> -+

>. \`r`nif`

(a) y>8 :`x, wkoro ` 5 < ir y>8 :`x,

wkro 6 < (st`ej`rj oqu`taie)

At as syhhotram `niut x -`xas,

[I

x8`

C[

F`tus romtuh

FQ

Q

y 8 :`x, 5

Q

Q

I[

F

C

F

x8`

[

F`tus romtuh

wkoro I as tko vortox,

C as tko cimus, FFas tko f`tus romtuh 2

['[7['['FFU noaed tko

`xas ic tko p`r`nif`.

(aa) x>8 :ny, wkoro n 5 < ir x>8 :ny,

wkoro n 6

>

>

>

8+ roprosoets `e offapso watk moetro `t (

8/10/2019 8_Areas of Bounded Regions

3/6

@^O@] IC NI_EJOJ ODAIE]?

www.tkaegaat.ae

>

0

>

>

xx>

>

xsae>

>

x:x

+-

+

-8 -

8 > sae0(0) : + > 8 (p >) sq. ueats

Ox.: Caej tko ro` ic tko marmfo :x>

+ :y>

8 1 wkamk as aetorair ti tko p`r`nif` x>

8 :y.]if. Davoe oqu`taies ti tko murvos ro

x>+ y>8

:

1...(0)

x>8 :y ...(>)

Oqu`taie (0) roprosoets marmfo wkiso moetro as () roprosoets p`r`nif` wkiso vortox as I(

\ut y 8:

x>ae (0), wo k`vo

:

1

03

xx

:> 8+

[

Q

[

Q

I

x 8:y>

)>/0,>(@ )>/0,>(N

> >

:x + :y 8 1> >

x:+ 03x> =3 8 + 0;) (x> >) 8 < x>8 > ir 0;

x 8 >

(Px>m`eeit no 0;)

V:

xy

>

8 8>

0

Z`gaed y 8

>

x:

1

- ie c(x) `ej y 8 :

x>

`s d(x), wkoro c(x) d (x) ae X>,>W wo k`vovo

8/10/2019 8_Areas of Bounded Regions

4/6

@^O@] IC NI_EJOJ ODAIE]4

www.tkaegaat.ae

Ox.3 Caej tko ro` ic tko rodaie niuejoj ny tko murvos y 8 x>+ >, y 8 x, x 8 < `ej x 8 =.

]if. Davoe oqu`taies ro

y 8 x> + > ...(0) y 8 x ...(>)

x 8 < ...(=) x 8 = ...(:)

Oqu`taie (0) roprosotes ` p`r`nif` watk vortox

@() `ej `xas as y-`xas.

Oqu`taie (>) roprosoets faeo p`ssaed tkriudk (

y 8 x

[

N(=,x( 8

=

=

x

x>=

x

-

+ 8

+-+

1

)31( 8 >

>0

sq. ueats.

Ox.4 Caej tko ro` ic tko rodaie

|(x,y) 2 x>+ y> >`x, y> >`x, ` 5 /=0>>> x=

`>

`

`xsae`

>

0)`x(`

-

-+--8 -

@( , )

M[

Q

[

Q

--

-8 -- )0(sae`

>

0`

=

>)

0 0>>0>

p--8

>`

>

0`

=

> >>8

-

p >>

`=

>

:

`sq. ueats.

Ox.; _saed tko hotkij ic aetodr taie, caej tko ro` ic tko tra`edfo @NM, miirjae`tos ic wkiso

vortamos `ro @ (>,

3

>

3

:jx

:

3x=jx)x1(jx

>

0,

>3

:

>:

>

>

x>>

x

:

=

>

xx1x>

>

x

>

?

--

-+

-

8 4 sq. ueats.

8/10/2019 8_Areas of Bounded Regions

5/6

@^O@] IC NI_EJOJ ODAIE]1

www.tkaegaat.ae

NI@^J \^INFO]

O[O^MA]O AA

P.0 _saed aetodr`taie, caej tko ro` ic tko rodaie, davoe 2

{(x, y) 2 < y x>+ 0 7 < y x + 0, < x >} WM.N.].O. >, 0) P as (=, :) `ej ^ as (?, >)WM.N.].O. > s`ej x 8 :. WM.N.].O. >y |x|}. WM.N.].O. >1}. WM.N.].O. >8 =>. M.N.].O. >8 :`y, wkoro 5 8+ `ej tko str`adkt faeo 0

=

y

:

x8+ WM.N.].O. >

8/10/2019 8_Areas of Bounded Regions

6/6

@^O@] IC NI_EJOJ ODAIE]00

www.tkaegaat.ae

@E]TO^ GOQ

O[O^MA]O A

0.>

0?>.

;

1=.

>

>0:.

3

4?.

>

`>

-

p0

>3.:p 4.

;

p;.

:

`np

>

`n

1. =

`03 >

0=

;

0>. =;

=:

-p

+p0=. =

=:03 +p0:. 1

0?. 03 03.

-

p>

0

:04.

=

00;.

3

>=01.

-

p

=

`>

:

` >>

>=>. : =.? :.

=

0?. 1/> 3.

>=

0+

;

1p

:

1sae

0=

0

4. :p ;. 03/= 1.=(p >) 0(:`n

-p

0>.=

>p

>

=0=.

-

p>

=

=

>0:.

=

:(;p = ) 0?.

=

03`

>03.

-

p=>

=

;

>0. 1 >>. (p >)