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snoitseuq hcae fo snoisivid owt yna TRAP ni -C

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in TRAP - dna B ni noisivid hcae rof skram)evif(5 TRAP -C ].

TRAP – A

(f fI .1 x si tahw neht noitcnuf ytisned ytilibaborp a si )

fo eulav eht

∫

𝑓

(

𝑥)

𝑥𝑑

∞

−

∞

?

21 era noitubirtsid laimonib a fo ecnairav dna naem eht fI .2

’p‘ dnif , 6 dna

‘ elbairav modnar a fI .3 X hcus noitubirtsid nossioP swollof ’ taht

𝑃(

𝑋

�

1)

�

𝑃

�

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2

� naem eht dnif ,

eht fo noitaived dradnats dna naem eht nwod etirW .4 noitubirtsid lamron dradnats

t5 = s fI .5 2 + t6 + yticolev laitini eht dnif ,5

evruc eht ot lamron fo epols eht dniF .6 y = x3 ,4( ta –2 )

o eht etatS .7 fo eerged dna redr

�

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2

�

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�

2

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0

:evloS .8

(

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2

�

94

)

𝑦

�

0

TRAP – B

ytilibaborp gniwollof eht sah ’X‘ elbairav modnar a fI .9

)X(E dnif , noitubirtsid


2

lamron fo seitreporp eerht yna noitneM .01 c evru

.11 fI x ea = t eb + –t lauqe syawla si noitarelecca eht taht wohS .

revo dessap ecnatsid eht ot

fo eulav muminim eht dniF .21 y = x 2 – 4x

:evloS .31

𝑥

𝑥𝑑

�

𝑦

𝑦𝑑

�

0

.41 fo rotcaf gnitargetni eht dniF

𝑦𝑑

𝑥𝑑

�

1

𝑥

𝑦

�

𝑥

.51 :evloS (

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2

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5

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�

6)

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�

0

fo largetni ralucitrap eht dniF .61 (

𝐷

2

�

01

𝐷

�

1)

𝑦

�

𝑒

−

𝑥

TRAP - C

)a( .71 elbairav modnar A ’X‘ gniwollof eht sah ytilibaborp

noitubirtsid

𝑋

0 1 2 3

𝑃

�

𝑋

�

𝑥

�

1

3

1

6

1

6

1

3

)i( dniF

𝐸(

𝑋) dna

�

𝑖𝑖

�

𝐸(

𝑋

2) )b( elbairav modnar A ’X‘ gniwollof eht sah ytilibaborp

noitubirtsid

X 0 1 2 3 4

)x = X( P a a3 a5 a7 a9

)i( dniF )ii( dna ’a‘

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�

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�

2

�

X 1 2 3

P (X)

1

2

0

1

2


3

)c( dna 51 = n fi ,noitubirtsid laimonib a nI

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�

1)

�

3

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�

0)

,

’p‘ fo eulav eht dnif fI )a( .81

%3 ,evitcefed era sblub cirtcele eht fo b 001 fo elpmas a ni taht ytilibaborp eht dnif sblu

( . evitcefed era sblub 5 yltcaxe

𝑒

−

3

�

0

.

8940

�

(b ) si naem noitubirtsid lamron a nI

01

dradnats dna

si noitaived

3

morf lavretni ytilibaborp eht dniF .

X ot 6.8 = X 8.21 =

)c( atad gniwollof eht rof enil thgiarts a tiF

)a( 91 nevig si elcitrap a fo noitauqe dellevart ecnatsid eht fI

yb noitarelecca eht taht wohS . t6 nis b + t6 soc a = s

ecnatsid sti sa seirav

)b( evruc eht ot stnegnat eht ot noitauqe eht dniF

y = x2 + x – ta 6 eht stuc ti erehw tniop eht x – sixa

fo seulav muminim dna mumixam eht dniF )c(

y 2 = x3 – 51 x2 – 63 x 81 +

esab fo enoc ralucric thgir a fo emulov eht dniF )a(.02

suidar

′

𝑟

′ thgieh dna

′

�

′ noitargetni gnisu yb

)b( :evloS

𝑛𝑎𝑡

𝑥

𝑒𝑠

𝑐

2

𝑦

𝑦𝑑

�

𝑛𝑎𝑡

𝑦

𝑒𝑠

𝑐

2

𝑥

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�

0

:evloS )c(

𝑦𝑑

𝑥𝑑

�

2

𝑦

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�

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2

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𝑥

:evloS )a( .12 (

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2)

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0

:evloS )b( (

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2

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8

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61 )

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:evloS )c( (

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2

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�

𝑛𝑖𝑠

9

𝑥

𝑥 0 1 2 3 4

𝑦 01 41 91 62 13


4

TRAP - A

(f fI .1 x si tahw neht noitcnuf ytisned ytilibaborp a si )

fo eulav eht

∫

𝒇

(

𝒙)

𝒙𝒅

∞

−

∞

?

ehT

eulav

fo

∫

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(

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�

1

∞

−

∞

.2 21 era noitubirtsid laimonib a fo ecnairav dna naem eht fI

’p‘ dnif , 6 dna

neviG : naeM =

𝑝𝑛 21 = - - - - - )1(

= ecnairaV

𝑞𝑝𝑛 6 = - - - - - )2(

�

2

�

�

1

�

⇒

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𝑝𝑛

�

6

21

⇒

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�

1

2

∴

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�

1

�

𝑞

�

1

�

1

2

�

𝟏

𝟐

3. ‘ elbairav modnar a fI X swollof ’ p hcus noitubirtsid nossio

taht

𝑷(

𝑿

�

𝟏)

�

𝑷

�

𝑿

�

𝟐

� naem dnif ,

:alumroF

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𝑋

�

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�

𝑒

−

𝜆

𝜆

𝑥

𝑥

!

neviG

𝑃(

𝑋

�

1)

�

𝑃

�

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�

2

�

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−

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1

1

!

�

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−

𝜆

𝜆

2

2

!

𝜆

1

�

𝜆

2

2

2

1

�

𝜆

2

𝜆

⇒

𝜆

�

2

∴ naeM

�

2

SREWSNA

rewsnA

rewsnA

rewsnA


5

4. etirW nwod eht naem fo noitaived dradnats dna

dradnats eht n noitubirtsid lamro

naeM

𝜇

�

0

noitaiveD dradnatS

𝜎

�

1

5. t5 = s fI 2 + t6 + yticolev laitini eht dnif ,5

:neviG t5 = s 2 + t6 + 5

v

�

𝑠𝑑

𝑡𝑑 = 5 ( t2 ) )1(6 + + 0 = 6 + t01

yticolev laitinI

v

� �

𝑠𝑑

𝑡𝑑 �

𝑡

=

0 = + )0(01 6 = ces / stinu 6

6 . fo epols eht dniF eht evruc eht ot lamron y = x3 ( ta 4 , –2)

:neviG y = x3

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�

3

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2

�

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𝑥𝑑�

�

4

,

−

2

�

�

3(

4)

2

�

84

,tnegnat eht fo epolS m 84 =

,lamron eht fo epolS

−

1

𝑚

�

−

1

84

7 . fo eerged dna redro eht etatS

�

𝒅

𝒚

𝒅

𝒙�

𝟐

�

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redrO

�

2

dna

eergeD

�

1

.8 :evloS

�

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�

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�

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�

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neviG (

𝐷

2

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)

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�

0

si noitauqe yrailixuA

𝑚

2

�

94

�

0

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2

�

94

rewsnA

rewsnA

rewsnA

rewsnA

rewsnA


6

𝑚

�

� √

94

�

�

7

∴ si noitulos ehT

𝑦

�

𝐴

𝑒

𝑚

1

𝑥

�

𝐵

𝑒

𝑚

2

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�

𝐴

𝑒

7

𝑥

�

𝐵

𝑒

−

7

𝑥

TRAP - B

.9 ytilibaborp gniwollof eht sah ’X‘ elbairav modnar a fI

d noitubirtsi f , )X(E dni

:alumroF

𝐸

�

𝑋

�

�

∑

𝑥

i

p

i

n

i

=

1

𝐸

�

𝑋

� =

𝑥

1

𝑝

1

�

𝑥

2

𝑝

2

�

⋯

�

𝑥

𝑛

𝑝

𝑛

= �

1

�

1

2�

�

(

2

�

0)

� �

3

�

1

2�

=

1

2

�

0

�

3

2

=

1

+

3

2

=

4

2

=

2

.01 noitneM yna eerht fo seitreporp n lamro c evru )i( . depahs lleb si evruc lamron ehT

)ii( . enil eht tuoba lacirtemmys si tI

𝑋

�

𝜇

.)iii( = edoM = naideM = naeM

𝜇

.11 fI s ea = t eb + –t . wohS eht taht syawla si noitarelecca

lauqe ecnatsid eht ot revo dessap

s :neviG = ea t eb + –t

X 1 2 3

P (X)

𝟏

𝟐

𝟎

𝟏

𝟐

rewsnA

rewsnA

rewsnA


7

v

�

𝑠𝑑

𝑡𝑑 ea = t – eb –t { ecnis

𝑑

𝑡𝑑(

𝑒

−

𝑡)

�

�

𝑒

−

𝑡 }

𝑎

�

𝑑

2

𝑠

𝑑

𝑡

2 ea = t eb + –t

⇒ a s =

∴

𝐴 noitarelecc revo dessap ecnatsid eht ot lauqe syawla si

.21 fo eulav muminim eht dniF y = x 2 – 4x

:neviG y = x 2 – 4x

y1 = 2x – 4

y2 = 2

tuP y 1 0 =

⇒ 2x – 0 = 4

2x 4 =

⇒ x = 2

woN (y2 ) x = 2 = 2 0 >

y si muminim ta x 2 =

ehT muminim fo eulav y )2( = 2 – )2(4

4 = – = 8 – 4

.31 evloS :

𝒙𝒅𝒙

�

𝒚𝒅𝒚

�

𝟎 G nevi

𝑥𝑑𝑥

�

𝑦𝑑𝑦

�

0

𝑥𝑑𝑥

�

�

𝑦𝑑𝑦

�

𝑥

𝑥𝑑

�

� �

𝑦

𝑦𝑑

𝑥

2

2

�

�

𝑦

2

2

�

𝐶

𝑥

2

2

�

𝑦

2

2

�

𝐶

rewsnA

rewsnA


8

.41 fo rotcaf gnitargetni eht dniF

𝒚𝒅

𝒙𝒅

�

𝟏

𝒙

𝒚

�

𝒙

neviG

𝑦𝑑

𝑥𝑑

�

1

𝑥

𝑦

�

𝑥

mrof eht fo si sihT

𝑦𝑑

𝑥𝑑

�

𝑦𝑃

�

𝑄

,ereH

𝑃

�

1

𝑥

;

𝑄

�

𝑥

rotcaF gnitargetnI

�

𝑒∫

𝑥𝑑𝑃

�

𝑒∫

1

𝑥

𝑥𝑑

�

𝑒

gol

𝑥

�

𝑥 15. evloS : �

𝑫

𝟐

�

𝑫𝟓

�

𝟔�

𝒚

�

𝟎

neviG (

𝐷

2

�

5

𝐷

�

6)

𝑦

�

0

uqe yrailixuA a si noit

𝑚

2

�

5

𝑚

�

6

�

0

(

𝑚

�

2)(

𝑚

�

3)

�

0

𝑚

�

2

�

0

𝑚

�

3

�

0

𝑚

�

2

𝑚

�

3

∴ si noitulos ehT

𝑦

�

𝐴

𝑒

𝑚

1

𝑥

�

𝐵

𝑒

𝑚

2

𝑥

𝑦

�

𝐴

𝑒

2

𝑥

�

𝐵

𝑒

3

𝑥

16 . eht dniF p ralucitra i fo largetn

�

𝑫

𝟐

�

𝟎𝟏

𝑫

�

𝟏�

𝒚

�

𝒆

−

𝒙

neviG (

𝐷

2

�

01

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1)

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�

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−

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𝑃

.

𝐼

.

�

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−

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2

−

01

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+

1

ecalpeR

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�

1

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.

𝐼

.

�

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−

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1

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2

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rewsnA


9

𝑃

.

𝐼

.

�

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−

𝑥

1

�

01

�

1

∴

,

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.

𝐼

.

�

𝑒

−

𝑥

21

TRAP - C

(.71 a .) elbairav modnar A ’X‘ gniwollof eht sah

ytilibaborp noitubirtsid

𝑿

𝟎 1 2 3

𝑷

�

𝑿

�

𝟏

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𝟔

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)i( dniF

𝑬(

𝑿) dna

�

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�

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)i( :alumroF

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�

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�

�

∑

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i

p

i

n

i

=

1

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1

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1

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2

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2

�

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1

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1

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6

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6

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6

6

=

9

6

�

3

2

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�

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i

2

n

i

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1

P

i

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2

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1

2

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1

�

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2

2

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2

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2

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0

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1

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1

2

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1

6�

�

�

2

2

�

1

6�

�

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3

2

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1

3�

= �

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1

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� �

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�

1

6�

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1

6�

�

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�

1

3�

rewsnA


01

=

0

�

1

6

�

4

6

�

3

=

1

+

4

+

81

6

=

32

6

.)b(.71 elbairav modnar A ’X‘ gniwollof eht sah

p ytilibabor noitubirtsid

X 0 1 2 3 4

P (X) a 3a 5a a7 9a

ii( dna ’a‘ )i( dniF )

𝑷

�

𝑿

�

𝟐

�

)i( taht wonk eW

∑

𝑷

𝒊 1 =

a + 3a + 5a + a7 + 9 a 1 =

52 = a 1

⇒52

1a

)ii(

𝑃

�

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�

2

� = P (X = 2 + ) P (X = 3) + P (X = 4)

= 7+ a5 + a 9a

= 12 a

= 1252

1

5212

.)c(.71 noitubirtsid laimonib a nI fi , = n 51 dna

𝑷(

𝑿

�

𝟏)

�

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𝑷(

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�

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,

dnif fo eulav eht ‘p’

neviG

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�

51

si noitubirtsid laimoniB

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(

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�

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�

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−

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(

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51

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51

−

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11

neviG

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(

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�

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�

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�

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51

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1

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1

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51

−

1

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3

51

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0

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0

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51

−

0

51

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41

�

3

�

1

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1

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51

51

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41

�

3

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51

51

𝑝

�

3

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51

𝑝

�

3(

1

�

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∴

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�

1

�

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51

𝑝

�

3

�

3

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51

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3

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3

81

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3

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3

81

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�

1

6

.)a(.81 fI

𝟑

% ,evitcefed era sblub cirtcele eht fo

sblub 001 fo elpmas a ni taht ytilibaborp eht dnif

5 yltcaxe . evitcefed era sblub (

𝒆

−

𝟑

�

𝟎

.

𝟖𝟗𝟒𝟎

�

:alumroF

𝑃(

𝑋

�

𝑥)

�

𝑒

−

𝜆

𝜆

𝑥

𝑥

!

neviG

𝑃

�

%3

�

3

001

;

𝑛

�

001

taht wonk eW

𝜆

�

𝑃𝑛

�

001 �

3

001�

�

3

𝑃(

𝑋

�

𝑥)

�

𝑒

−

3(

3)

𝑥

𝑥

!

:evitcefed era 5 yltcaxE

𝑃

�

𝑋

�

5

�

𝑃(

𝑋

�

5)

�

𝑒

−

3(

3)

5

5

!

𝑃(

𝑋

�

5)

�

0

.

8940

�

342

�

021

�

0

.

8001

rewsnA


21

.)b(.81 si naem noitubirtsid lamron a fI 𝟎𝟏 dradnats dna

si noitaived 𝟑 . morf lavretni ytilibaborp eht dniF

X = ot 6.8 X 8.21 =

,neviG naeM 𝜇 = 01

noitaiveD dradnatS 𝜎 = 3

taht wonk eW 𝑧 = 𝑋−𝜇𝜎

= 𝑋 − 01

3

P morf lavretni ytilibabor X = ot 6.8 X 8.21 =

𝑷( 𝟖. 𝟔 < 𝑋 < 𝟐𝟏 . 𝟖 )

nehW 𝑋 = 8.6 nehW 𝑋 = 21 .8

𝑧 =8.6 − 01

3 𝑧 =

21 .8 − 013

𝑧 =−1.4

3 𝑧 =

2.83

𝑧 = −0. 664 = −0. 74 𝑧 = 0. 339 = 0. 39

∴ 𝑷( 𝟖. 𝟔 < 𝑋 < 𝟐𝟏 . 𝟖 ) = 𝑷(−𝟎. 𝟕𝟒 < 𝑧 < 0. 39 )

−∞ 𝑧 = −0. 74 0 𝑧 = 0. 39 ∞

= 𝑃(−0. 74 < 𝑧 < 0) + 𝑃(0 < 𝑧 < 0. 39 )

= 𝑃(0 < 𝑧 < 0. 74 ) + 𝑃(0 < 𝑧 < 0. 39 )

= 0. 8081 + 0. 8323

= 𝟎. 𝟔𝟒𝟎𝟓

rewsnA


31

)c(.81 atad gniwollof eht rof enil thgiarts a tiF

𝒙 0 1 2 3 4

𝒚 01 41 91 62 13

teL

𝑦

�

𝑥𝑎

�

𝑏

…

�

1

� tif tseb fo enil eht eb era snoitauqe lamron eht nehT

𝑎 ∑

𝑥

𝑖

�

𝑏𝑛

� ∑

𝑦

𝑖 .... )2(

𝑎 ∑

𝑥

𝑖

2

�

𝑏 ∑

𝑥

𝑖

� ∑

𝑥

𝑖

𝑦

𝑖 .... )3( etupmoc eW ∑

𝑖𝑥

, ∑

𝑥

𝑖

2

, ∑

𝑖𝑦

dna

∑

𝑖𝑥

𝑖𝑦

.elbat gniwollof eht morf

𝑖𝑥

𝑖𝑦

𝑖𝑥

2

𝑖𝑦𝑖𝑥 0 01 0 0 1 41 1 14 2 91 4 83 3 62 9 87 4 13 61 1 42

∑

𝒙

𝒊

�

01 ∑

𝒚

𝒊

�

001 ∑

𝒙

𝒊

𝟐

�

03 ∑

𝒙

𝒊

𝒚

𝒊

�

452 ,ereH

𝑛

�

5

teg ew ,snoitauqe lamron eht gnisU

)2(

⟹

𝒂 ∑

𝒙

𝒊

�

𝒃𝒏

� ∑

𝒚

𝒊

𝑎(

01 )

�

�

5

�

𝑏

�

001

01

𝑎

�

5

𝑏

�

001

)3(

⟹

𝑎 ∑

𝑥

𝑖

2

�

𝑏 ∑

𝑥

𝑖

� ∑

𝑥

𝑖

𝑦

𝑖

𝑎(

03 )

�

𝑏

�

01

�

�

452

03

𝑎

�

01

𝑏

�

452

: eluR s’remarC yB

01

𝑎

�

5

𝑏

�

001 dna

03

𝑎

�

01

𝑏

�

452

�

�

�

01

5

03

01 �

�

001

�

051

�

�

05

rewsnA


41

�

𝑎

�

�

001

5

452

01 �

�

0001

�

0721

�

�

072

�

𝑏

�

�

01

001

03

452 �

�

0452

�

0003

�

�

064

𝑎

�

�

𝑎

�

�

�

072

�

05

�

𝟓

.

𝟒

,

𝑏

�

�

𝑏

�

�

�

064

�

05

�

𝟗

.

𝟐

)1(

⇒

𝑦

�

𝑥𝑎

�

𝑏

tuP

𝑎

�

5

.

4 dna

𝑏

�

9

.

2

𝒚

�

𝟓

.

𝟒

𝒙

�

𝟗

.

𝟐 .tif tseb fo enil eht si hcihw ,

�.91 .)a eht fI dellevart ecnatsid fo noitauqe nevig si elcitrap a

yb a = s soc + t6 b t6 nis . noitarelecca eht taht wohS

sa seirav ecnatsid sti

s t6 nis b + t6 soc a = –––– )1(

v

�

𝑠𝑑

𝑡𝑑

v a = { (– )t6nis 6 } b + { )t6 soc( 6}

v = – t6 soc b6 + t6nis a6

a

�

𝑑

2

𝑠

𝑑

𝑡

2

a = – a6 { )t6soc( 6 } b6 + {(– 6 )t6 nis }

a = – t6soc a63 – t6nis b63

a = – 63 { t6 nisb + t6soca }

a = – 63 {s} [ gnisu ])1(

a = – 63 { ecnatsiD }

∴

.ecnatsid sti sa seirav noitarelecca

.)b(.91 noitauqe eht dniF fo stnegnat eht eht ot evruc

y = x2 + x – 6 ti erehw tniop eht ta eht stuc x – sixa

rewsnA


51

y = x2 + x – 6

𝑦𝑑

𝑥𝑑

�

2

𝑥

�

1

evruc ehT y = x2 + x – 6 stuc x – sixa . tuP 0 = y

x2 + x – 6 0 = (x – 2 ( ) x + 3 0 = )

x – 0 = 2 x + 3 0 = x 2 = x = –3

( dna )0 ,2( – )0 ,3

)i( )0 ,2( ta tnegnat eht fo epolS

�

𝑦𝑑

𝑥𝑑�

�

2

,

0

�

�

2(

2)

�

1

�

5

�

𝑚

fo noitauqE eht ( ta tnegnat si )0 ,2 y – y1 = m ( x – x1)

ereH m = 5 , x1 ,2 = y1 0 =

y – = 0 5 (x –2)

y = 5x – 01

5x – y – 0 = 01

�

𝒊𝒊

�

( ta tnegnat eht fo epolS – )0 ,3

�

𝑦𝑑

𝑥𝑑�

�

−

3

,

0

�

�

2(

�

3)

�

1

�

�

5

�

𝑚

fo noitauqE eht ( ta tnegnat –3 si )0 , y – y1 = m (x – x1) ereH m = –5 , x1 = –3 , y1 0 =

y – = 0 –5 (x – (–3))

y = – 5 (x + )3

y = –5x – 51

5x + y 0 = 51 +

rewsnA


61

.)c(.91 eht dniF fo seulav muminim dna mumixam

y = 2x3 – 51 x2 – 63 x 81 +

y 2 = x3 – 51 x2 – 63 x 81 +

y 1 2 = (3x2) – 51 (2x) – 63 (1) 0+

y1 6 = x2 – 03 x – 63

y2 6 = (2x) – 03 (1) – 0

y2 21 = x – 03

tuP y 1 0 =

⇒ 6x2 – 03 x – 0 = 63

nO teg ew ,6 yb x2 – 5x – 0 = 6

( x 1 + ( ) x – 6 0 = )

x + 1 0 = x – 6 0 =

x = – 1 x 6 =

esaC )i( : nehW x = – 1

[ woN y2] x =-1 (21 = – 1 ) – 03

= –1 2 – 03 = – 24 0 <

y ta mumixam si x = – 1

fo eulav mumixam ehT y (2 = – 1)3 – (51 – 1)2 – (63 – 1 81 + )

= – 2 – + 51 63 81 +

= 73

)ii( esaC : nehW x = 6

[ woN y2] x = 6 (21 = 6 ) – 03

= 27 – 03 = 24 0 >

rewsnA


71

y ta muminim si x = 6

(2 = y fo eulav muminim ehT 6)3 – (51 6) 2 – 6(63 ) 81 +

(2 = 612 ) – (51 63 ) – 612 81 +

4 = 23 – 045 – 612 81 +

= – 603

.)a(.02 esab fo enoc ralucric thgir a fo emulov eht dniF

suidar ′𝒓′ thgieh dna ′𝒉′ noitargetni yb

thgir a gnitator yb demrof si enoc ralucric thgir A

tuoba elgnairt delgna 𝑥- sixa .

Y

𝒚 = 𝒙𝒎 r

𝑿′ x ) 0= 𝜃 ℎ M 𝑿

𝒀′

era stimil ehT 𝑥 = 0 dna 𝑥 = ℎ

∴ 𝑎 = 0 𝑑𝑛𝑎 𝑏 = ℎ

enil eht fo noitauqE 𝑨𝑶 si 𝒚 = 𝒙𝒎 … … …. (𝟏)

nI △ 𝑀𝐴𝑂 , nat 𝜃 =𝑝𝑝𝑜 . 𝑒𝑑𝑖𝑠𝑗𝑑𝑎 . 𝑒𝑑𝑖𝑠

rewsnA


81

nat

𝜃

�

𝑟

�

⇒

𝑚

�

𝑟

ℎ ecnis

𝑚

�

nat

𝜃

(

1)

⇒

𝑦

� �

𝑟

��

𝑥

𝑒𝑚𝑢𝑙𝑜𝑉

𝑓𝑜

𝑒𝑛𝑜𝑐

�

𝜋 ∫

𝑦

2

𝑏

𝑎

𝑥𝑑

�

𝜋 � �

𝑥𝑟

� �

2

ℎ

0

𝑥𝑑

�

𝜋 �

𝑟

2

𝑥

2

�

2

ℎ

0

𝑥𝑑

�

𝜋

𝑟

2

�

2 �

𝑥

2

ℎ

0

𝑥𝑑

�

𝜋

𝑟

2

�

2 �

𝑥

3

3�

0

ℎ

�

𝜋

𝑟

2

�

2 �

�

3

3 �

�

𝜋

𝑟

2

ℎ

3

𝑐𝑖𝑏𝑢𝑐

𝑠𝑡𝑖𝑛𝑢

.)b(.02 evloS :

𝒏𝒂𝒕

𝒙

𝒆𝒔

𝒄

𝟐

𝒚

𝒚𝒅

�

𝒏𝒂𝒕

𝒚

𝒆𝒔

𝒄

𝟐

𝒙

𝒙𝒅

�

𝟎

neviG

𝑛𝑎𝑡

𝑥

𝑒𝑠

𝑐

2

𝑦

𝑦𝑑

�

𝑛𝑎𝑡

𝑦

𝑒𝑠

𝑐

2

𝑥

𝑥𝑑

�

0

nat

𝑥

ces

2

𝑦

𝑦𝑑

�

�

nat

𝑦

ces

2

𝑥

𝑥𝑑

rewsnA


91

ces

2

𝑦

nat

𝑦

𝑦𝑑

�

�

ces

2

𝑥

nat

𝑥

𝑥𝑑

�

ces

2

𝑦

nat

𝑦

𝑦𝑑

�

� �

ces

2

𝑥

nat

𝑥

𝑥𝑑

gol (

nat

𝑦)

�

�

gol

(

nat

𝑥)

�

gol

𝐶

gol (

nat

𝑦)

�

gol

(

nat

𝑥)

�

gol

𝐶

gol (

nat

𝑦

nat

𝑥)

�

gol

𝐶

nat

𝑦

nat

𝑥

�

𝐶

(.02 .)c evloS :

𝒚𝒅

𝒙𝒅

�

𝒚𝟐

𝒙

�

𝒙

𝟐

𝒏𝒊𝒔

𝒙

neviG

𝑦𝑑

𝑥𝑑

�

2

𝑦

𝑥

�

𝑥

2

nis

𝑥

mrof eht fo si sihT

𝑦𝑑

𝑥𝑑

�

𝑦𝑃

�

𝑄

ereH

𝑃

�

�

2

𝑥

,

𝑄

�

𝑥

2

nis

𝑥

rotcaF gnitargetnI

�

𝑒∫

𝑥𝑑𝑃

�

𝑒∫

−

2

𝑥

𝑥𝑑

�

𝑒

−

2 ∫

1

𝑥

𝑥𝑑

�

𝑒

−

2

gol

𝑥

�

𝑒

gol

𝑥

−

2

�

𝑥

−

2

�

1

𝑥

2

∴

, si noitulos ehT

𝑦

𝑒∫

𝑥𝑑𝑃

� ∫

𝑄

𝑒∫

𝑃

𝑥𝑑

𝑥𝑑

�

C

𝑦

1

𝑥

2

� �

𝑥

2

nis

𝑥 �

1

𝑥

2�

𝑥𝑑

�

C

� �

nis

𝑥

𝑥𝑑

�

C

�

�

soc

𝑥

�

C

rewsnA


02

.)a(.12 evloS : �

𝑫

𝟐

�

𝑫

�

𝟐�

𝒚

�

𝟎

neviG (

𝐷

2

�

𝐷

�

2)

𝑦

�

0

uqe yrailixuA a si noit

𝑚

2

�

𝑚

�

2

�

0

(

𝑚

�

1)(

𝑚

�

2)

�

0

(

𝑚

�

1)

�

0

(

𝑚

�

2)

�

0

𝑚

�

1

𝑚

�

�

2

∴

si noitulos ehT

𝑦

�

𝐴

𝑒

𝑚

1

𝑥

�

𝐵

𝑒

𝑚

2

𝑥

𝑦

�

𝐴

𝑒

𝑥

�

𝐵

𝑒

−

2

𝑥

.)b(.12 evloS : �

𝑫

𝟐

�

𝑫𝟖

�

𝟔𝟏 �

𝒚

�

𝒆𝟐

𝒙

,neviG (

𝐷

2

�

8

𝐷

�

61 )

𝑦

�

2

𝑒

𝑥

si noitauqE yrailixuA

𝑚

2

�

8

𝑚

�

61

�

0

(

𝑚

�

4)(

𝑚

�

4)

�

0

𝑚

�

4

�

0

𝑚

�

4

�

0

𝑚

�

4

𝑚

�

4

∴

, noitcnuF yratnemelpmoC

�

�

𝒙𝑨

�

𝑩

�

𝒆

𝒙𝒎

(

𝐶

.

𝐹)

� (

𝑥𝐴

�

𝐵)

𝑒

4

𝑥

largetnI ralucitraP

�

𝑒

𝑥𝑎

𝑓(

𝐷)

𝑃

.

𝐼

.

�

2

𝑒

𝑥

𝐷

2

−

8

𝐷

+

61

ecalpeR

𝐷

𝑦𝑏

1

𝑃

.

𝐼

.

�

2

𝑒

𝑥

�

1

�

2

�

8

�

1

�

�

61

𝑃

.

𝐼

.

�

2

𝑒

𝑥

1

�

8

�

61

rewsnA

rewsnA


12

𝑃

.

𝐼

.

�

2

𝑒

𝑥

9

,noituloS lareneG

𝑦

�

𝐶

.

𝐹

�

𝑃

.

𝐼

𝑦

� (

𝑥𝐴

�

𝐵)

𝑒

4

𝑥

�

2

𝑒

𝑥

9

�.12 .)c evloS : �

𝑫

𝟐

�

𝟔𝟏 �

𝒚

�

𝒏𝒊𝒔

𝒙𝟗

neviG

(

𝐷

2

�

61 )

𝑦

�

𝑛𝑖𝑠

9

𝑥

si noitauqe yrailixuA

𝑚

2

�

61

�

0

𝑚

2

�

�

61

𝑚

�

� √

�

61

𝑚

�

�

𝑖

4

,ereH

𝛼

�

0

,

𝛽

�

4

∴ noitcnuf yratnemelpmoC

�

𝒆

𝒙𝜶

�

𝑨

𝒔𝒐𝒄

𝒙𝜷

�

𝒏𝒊𝒔𝑩

𝒙𝜷

�

�

𝐶

.

𝐹

�

𝑒

0

𝑥

�

𝑠𝑜𝑐𝐴

4

𝑥

�

𝑛𝑖𝑠𝐵

4

𝑥

�

�

𝑠𝑜𝑐𝐴

4

𝑥

�

𝑛𝑖𝑠𝐵

4

𝑥

𝑟𝑎𝑙𝑢𝑐𝑖𝑡𝑟𝑎𝑃

𝑙𝑎𝑟𝑔𝑒𝑡𝑛𝐼

�

nis

𝑥𝑎

𝑓

�

𝐷

�

�

nis

9

𝑥

𝐷

2

�

61

ecalpeR

𝐷

2

�

� [

9]

2

�

�

18

�

nis

9

𝑥

�

18

�

61

�

nis

9

𝑥

�

56

si noitulos lareneG

𝑦

�

𝐶

.

𝐹

�

𝑃

.

𝐼

∴

𝑦

�

𝑠𝑜𝑐𝐴

4

𝑥

�

𝑛𝑖𝑠𝐵

4

𝑥

�

nis

9

𝑥

56

rewsnA

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