Core 2 - Ch 11 - 1 - Integration - Solutions

8/13/2019 Core 2 - Ch 11 - 1 - Integration - Solutions

1/22

1. (a)c

xxxx

xx +++=+

23d)

243(

4

2

3

M1 A2 (1, 0) 3

(b)

2

1

4

2

3

2

1

23d)

243(

++=+

x

xxxx

x

= (6 + 16 + 1) (3 + 1 + 2) M1

= 17 A1 2[5]

2. (a) y= 9 8 42

= 0 b= 4 (*) B1 c.s.o. 1

(b) x

y

d

d

= 2 + 23

x M1

Whenx= 1 gad!en" = 2 + 1 = 1 A1#o e$%a"!on o& "he "angen" !sy ' = 1 (x 1) M1

!.e. y+x= 6 (*) A1 c.s.o. 4

(c) e"y= 0 andx= 6 soD!s (6, 0) B1 1

(d) Aea o& "!ange = 21

* ' * ' = 12.' B1

4

1 (9 2x2 21

x ) dx=

4

121

22

1

29

xxx

Ignore limits M1 A1

= (36 16 4 * 2) (9 1 4)

Use of limits M1= 12 4

= 8 A1

#o shaded aea !s 12.' 8 = 4.' A1 6[12]

3. (a) Ay= 1 By= 4 B1

(b) 2'

2

d

d x

x

y == '

2

heex= ' M1 A1

-angen" y1 = '

2

(x') ('y= 2x') M1 A1

(c) x=2

1

'y B1 B1

(d) n"ega"e

=

3

10

23

' 23

23

yy

M1 A1 &"

/

4

/ 1=

3

110

3

410 23

23

, =3

70

(233

1

, 23.3) M1 A1, A1

Holland Park School 1


2/22

A"ena"!e &o (d) n"ega"e 7'

3x

M1 A1

Aea = (10 * 4) (' * 1)

7'

12'

7'

1000

, = 3

70

(233

1

, 23.3) M1 A1, A1

n bo"h (d) schees, &!na M !s scoed %s!ng cand!da"es 45 and 15. [12]

4. (a) x2 2X+ 3 = 9 x M1

x2x 6 = 0 (x+ 2)(x3) = 0 x= 2, 3 M1 A1

y= 11, 6 M1 A1 &" '

(b)( ) xxxxxx 3

3d32 2

32 +=+

M1 A1

( )

+=

+

64

3

89993

3

3

2

23

xxx

= 32

21 M1 A1

-ae!% 2

1

(11 + 6) * '

=

2

142

B1 &"

Aea = 6

'20

3

221

2

142 =

M1 A1 7

A"ena"!e (9 x) (x2 2x+ 3) = 6 +xx

2M1 A1

( )32

6d632

2 xxxxxx +=+M1 A1 &"

6'20,

382129

2918

326

3

2

32

=

++

+=

+

xxx

M1 A1, A1[12]



3/22

5. (a) 4x+ 9, +12 x (Ao 6x+ 6x) B1, B1 2Ao (2x)2o 4x2on !& a"e o :%s"!&!es %ndes"and!ng.

(b) ++=++ xxxdxxx 982)9124( 2

322

1

(&" de. on 3 "es) M1 A1 &"

)18)28(8(/..... 23

21 ++= (2 + 8 + 9) M1

222 23

= (seen o !!ed) B1

= 7 + 16 2 A1 '

#ec!a case M!sead as 2(x+ 3)

(a) B1 &o 4x+ 12

(b) ;%s" "he "o M as ae aa!abe.[7]

6. (a)


4/22

(d) 2

'

3

6

4d)'6(

23423 xxxxxxx +=+

M1 A1

a%a"!ng a" one o& "he!xa%e

=+

4

3

2

'2

4

1

M1 A1 &"

a%a"!ng a" "he o"hexa%e

=+ 41

312

12'

2'04

62'

A1

/>..' />..

1o />..

1 />..

'M1

31 4

3

4

1 = 32

-o"a Aea = 32 + 4

3

= 324

3

A1 7

If integrating t#e wrong ex!ression in (d)$ (e.g. x2% &x ' )$

do not allow t#e first M mar$ b"t t#en follow sc#eme.[14]

7. (a) #oe

32

4

1

2

3xx

= 0 "o &!nd!= 6, o e!&

32 64

16

2

3 = 0 (?) B1 1

(b) 4

33

d

d 2xx

x

y=

M1 A1

m= 9,y 0 = 9(x 6) (An coec" &o) M1 A1 4

(c) 4

33

2xx

= 0,x= 4 M1, A1&" 2

(d) 162

d42

3 4332

xxx

xx =

(Ao %ns!!&!ed es!ons) M1 A1

[ ] 2716

6

2

6.........

436

0 == M @eed 6 and 0 as !!"s. M1 A1 4

[11]

8. (a) x

y

d

d

= 3x2 14x+ 1' M1 A1 2



5/22

(b) 3x2 14x+ 1' = 0 M1

(3x ')(x 3) = 0 x= >, M1, A1

(A re*"ires correct *"adratic factors).

y= 12 A1 4

(+ollowing from x , -)

(c) x= 1 y= 12 B1

#aeycood. as /(o eo gad!en"5), so/!s aae "o "hexa!s B1 2

(d)( ) xxxxxxxx 3

2

1'

3

7

4d31'7

23423 ++=++

M1 A1 A1

(+irst A - terms correct$ 1econd A all correct)

++

++=

++ 3

2

1'

3

7

4

19

2

13'63

4

813

2

1'

3

7

4

3

1

234

xxxx

M1

12

'8

4

333

24 = 2'3

1

(2 * 12) = 13

1

M1 A1 6

(or e*"i. or - s.f or better)[14]

9. (a) y= 4xx

2

x

y

d

d

= 4 2x M1 A154 2x5 = 2, x= > M1

x= 3,y= 3 A1 4

(b) xcood!na"e o&A!s 4 B1

( )

=

32

4d4

322 xxxxx

M1 A1&"

===

3

210

3

32

3

6432

32

44

0

32xx

(o eac" e$%!aen") M1 A1 '

[9]



6/22

10. (a) (x 3)2, +9 !s . a= 3 and b= 9 a :%s" be !""en

don !"h no e"hod shon. B1, M1 A1 3

(b) C !s (3, 9) B1

(c) A = (0, 18) B1

x

y

d

d

= 2x 6, a"A m= 6 M1 A1

$%a"!on o& "angen" !sy 18 = 6x (!n an &o) A1&" 4

(d) #ho!ng "ha" !ne ee"s a!s d!ec" beo C, !.e. a"x= 3. A1cso 1

(e) A= x2 6x+ 18x= /31

x3 3x

2+ 18x M1 A1

#%bs"!"%"!ngx= 3 "o &!nd aeaA%nde c%eA/=36 M1

Aea o&3=A aea o& "!ange =A 2

1

* 18 3, = 9 M1 A1 '

A"ena"!e x2 6x+ 18 (18 6x)dx M1

=31

3x

M1 A1 &"

Dsex= 3 "o g!e anse 9 M1 A1[13]

11. (a) x2+ 6x+ 10 = 3x+ 20 M1

x2+ 3x 10 = 0(x+ ')(x 2) = 0 sox=, ' o 2 M1, A1

s%b a a%e &ox"o ob"a!n a a%e &oy!ny= 3x+ 20,y= ' o 26 M1, A1 '

(b) !ne c%e =, 10 3xx2

M1, A1

32

310)310(

322 xxxdxxx =

M1 A2E1E0&.".

)3

12'2'

2

3'0()

3

84

2

320(

32

310

2

'

32 +=

xxx

M1

6

1'7

6

'4'

3

111 ==

A1 7

[12]



7/22

A- (b)xx

xdxxx 103

3)106( 2

32 ++=++

(1 each !ncoec" "e) M1 A2E1E0

%se o& !!"s = ( 38

+ 12 + 20) ( 3

12'

+ 7' '0) = ('1 31

) M1

Aea o& -ae!% = 2

1

(' + 26)(2 ') = (108 21 )

o 46 62.' (&o !n"ega"!on) B1&.".

#haded aea = -ae!% 6

13

12

1 '7'1108 == o 6343

o 61.'7

M1 A1 7

12. (a) x

y

d

d

= 6 + 8x3

M1 A1 2M is for x

nx

n%

in at least one term$ & or x

%-is s"fficient.

A is f"lly correct answer.

Ignore s"bse*"ent woring.

(b) 2

6d

2xxy =

+ 4x1

+ 4 M1 A1 A1 3

M 4orrect !ower of x in at least one term (4 s"fficient)

+irst A 2

&x 2

' 4

1econd A ' 5x%

[5]

13. (a)

21

3d

dx

x

y=

6 M1 A1

21

3x 6 = 0,2

1

x = 2 x= 4 (?) M1 A1 4

+irst M for decrease of in !ower of x of at least

one term (disa!!earance of 607 s"fficient)

1econd M for !"tting x

y

d

d

, 0 and finding x , 8.



8/22

(b)

+=

+ xx

xxxx 103

'

4d1062 2

2'

23

M1 A1 A1

( )

+

+=

+ 103

'

440163

'

44103

'

4 2'

4

1

22

'

xxx

M1 A1&"

(= 17.6 7.8 = 9.8)

F!nd!ng aea o& "ae!% = 2

1

(6 + 2) * 3 (=12) M1 A1

/A = (1, 6), B = (4, 2)

G b !n"ega"!on

4

1

2

3

222

xx

Aea o&3=12 9.8 = 2.2 A1 8

+irst M ower of at least one term increased by

+irst A +or '

4 2E'x

1econd A +or % -x2' 0x

1econd M for limits re*"ires 9 [ ] [ ]14 9 (allow candidate:s

657)

and some !rocessing of 6integral7$[ ] 4

1y is M0

Aft re*"ires 1and 4s"bstit"ted in candidate:s 3-termed

integrand("nsim!lified)

Area of tra!e;i"m M attem!t at < (yA

' yB)(x

B% )

orx

xd

3

422

(A correct "nsim!lified)

H-IA

A""e"!ng !n"ega J( e$%a"!on o& !ne e$%a"!on o& c%e)K -h!d M1

=xxx dJ)2

3

14

3

8(J 2

3

+Fo%"h A1

Ce&o!ng !n"ega"!on F!s" M1

+ )'4

()3

7

3

8

( 2

'

2

xxx

)'

4

(2

'

xF!s" A1

+ J)

3

7

3

8(J 2xx

ao as &oo "ho%gh !n "h!s case. #econd A1

!!"s M1A1 #econd M1

-h!d A1

Anse A1 F!&"h A1[12]

14. (a)x

xxxxx '

1

8d)'82(

122

+=+

M1 A1 A1



9/22

4

1

12 '

1

8

+

xx

x

= (16 2 20) (1 8 ') (= 6) M1

x= 1y= ' andx= 4y= 3.' B1

Aea o& "ae!% = 2

1

(' + 3.')(4 1) (= 12.7') M1

#haded aea = 12.7' 6 = 6.7' M1 A1 8(M 1"btract eit#er was ro"nd)

Integration =ne term wrong M A A0> two terms

wrong M A0 A0.

?imits M for s"bstit"ting limits 5 and into a c#anged

f"nction$ and s"btracting t#e rig#t way ro"nd.

Alternatie

x= 1y= ' andx= 4y= 3.' B1

$%a"!on o& !ney ' = 2

1

(x 1) y= 2

1

2

11

x, s%bse$%en"%sed !n !n"ega"!on !"h !!"s. 3

dM1

+

'

82

2

1

2

112

xxx

4"h

M1

(M 1"btract eit#er way ro"nd)

=

1

8

4

'

2

21d8

2

'

2

21 122 xxxxxx

1s M1 A1&" A1&"

(enalise integration mistaes$ not algebra

for t#e ft mars)

++=

84

'

2

21)22042(

1

8

4

'

2

214

1

12 xxx

2nd

M1

(M 3ig#t way ro"nd)

#haded aea = 6.7' A1

(@#e follow t#ro"g# mars are for t#e s"btracted ersion$

and again ded"ct an acc"racy mar for a wrong term

=ne wrong M A A0> two wrong M A0 A0.)



10/22

(b) x

y

d

d

= 2 16x3

M1 A1

(nceas!ng hee) x

y

d

d

L 0 FoxL 2,3

16

x N 2, xy

d

d

L 0 dM1 A1 4

(allow )

Alternatie for t#e last 2 mars in (b)

M1 #ho "ha"x= 2 !s a !n!%, %s!ng, e.g., 2nd

de!a"!e.

A1


11/22

(b)166

d

d2

2

= xx

y

A"x= 2,...

d

d2

2

=x

y

M1

4 (o N 0, o bo"h), "hee&oe a!% A1&" 2M Attem!t second differentiation and s"bstit"tion of one of

t#e x al"es.

Aft 3e*"ires correct second deriatie and negatie al"e of

t#e second deriatie$ b"t ft from t#eir x al"e.

(c)( ) ++=+ )(2

20

3

8

4d208

23423 4

xxxxxxx

M1 A1 A1 3

All - terms correct M A A$

@wo terms correct M A A0$

=ne !ower correct M A0 A0.

(d)40

3

644 +

=

3

68

M1

Ax= 2 y= 8 32 + 40 = 16 (Ma be scoed esehee) B1

Aea o& =162

3

10

2

1

AAB yxx )(

2

1

=

3

32

M1

#haded aea =

==+

3

133

3

100

3

32

3

68

M1 A1 '

?imits M 1"bstit"ting t#eir lower x al"e into a c#anged:ex!ression.

Area of triangle M +"lly correct met#od.

Alternatie for t#e triangle (finding an e*"ation for t#e straig#t

line t#en integrating) re*"ires a f"lly correct met#od to score

t#e M mar.

+inal M +"lly correct met#od (beware alid alternaties)

[14]

17. (a)

4184d

d += xxx

y

xn x

n1M1 A1 2



12/22

(b)( ) 2332 3

32d62

+= xxxxxx

n x

n+1M1 A1

[ ]

++= 3

32

933

32... 3

3

1

M1

= 1432

344

, 9132

o e$%!aen" A1 4[6]

18. (a) y= 0 21

x (3 x) = 0 x= 3 B1 1

o 33 23

3 = 33 33 = 0

(b)

21

21

23

23

dd xxxy =

xn x

n1M1 A1

21

21

0d

dxx

x

y==

Dse o& x

y

d

d

= 0 M1

x= 1 A1

A (1, 2) A1 '

(c)

2'

23

23

21

'

22d3 xxxxx =

M1xn xn+1 M1 A1+A1

Acce" %ns!!&!ed eess!ons &o As

Aea =[ ] 30... = 2 * 33 '

2

* 93 Dse o& coec" !!"s M1

Aea !s '12

3 (%n!"s2) A1 '

Fo &!na A1, "es %s" be coec"ed "oge"he b%" acce" eac"

e$%!aen"s, e.g.'4

27 [11]

19. (a) 2x + 4 = x23

M1

4x2 8x + 3 = 0 A1

(2x 3)(2x ) = 0 M1

x= 0.', 1.' A1 4



13/22

(b)[ ] 1)13(

21o442

'.1

'.0

'.1'.0

2 ++=+ xxdxxM1

= 2 A1

'.1

'.0

'.1

'.0n

23

23

= xdxx M1 A1

=3n

23

A1&"

Aea = 2 23

n 3 A1 6

A"ena"!e so%"!on

Aea =dx

xx +

'.1

'.0 2

342M1

'.1

'.0

2 n1234

+= xxx

M1 A1 A1

o.e.3n12326

41

49 ++=

A1&"

= 2 23

n 3 A1[10]

20. (a) 23

= 2x2+ 4x M1

4x2 8x + 3(= 0) A1

(2x 1)(2x 3) = 0 M1

23

21 ,=x

A1 4



14/22

(b) Aea o&3 =

23d)42(

23

21

2 + xxx(&o 2

3

) B1

( )

+=+ 232 232d42 xxxxx

(Ao O/ , acce" 2

4

x2

M1 /A1

+

+=+ 232

2

3

323

21

2

2

122

132,

2

322

332d)42( xxx

M1 M1

=

611

Aea o&3= 31

23

611 =

(Acce" eac" e$%!aen" b%" no" 0.33 ...) A1cao 6[10]

(a) 1s"M1 &o &o!ng a coec" e$%a"!on

1s"A1 &o a coec" 3-P (condone !ss!ng = 0 b%" %s" hae a "es on one s!de)

2nd

M1 &o a""e"!ng "o soe ao!a"e 3-P

(b) B1 &o s%b"ac"!on o& 23

. !"he c%e !ne5 o !n"ega ec"ange5

1s"M1 &o soe coec" a""e" a" !n"ega"!on (x

nxn+1)

1s"A1 &o 3

2

x3+ 2x

2on !.e. can !gnoe%2

3

x

2nd

M1 &o soe coec" %se o& "he! 23

as a !!" !n !n"ega

3d

M1 &o soe coec" %se o& "he! 21

as a !!" !n !n"ega and

s%b"ac"!on e!"he a o%nd

#ec!a


15/22

Acce"abe a"ena"!es !nc%de

3x2+ 6x

1 3x

2+ 3*2x 3x

2+ 6x + 0

gnoe Q# (e.g. %se /he"he coec" o no" o& x

y

d

d

and2

2

d

d

x

y

)

3x2+ 6x + c o 3x

2+ 6x + cons"an" (!.e. "he !""en od cons"an") !s B0 B1

M1 A""e" "o d!&&een"!a"e "he! &(x)xnRxn 1.x

nRx

n 1seen !n a" eas" one o& "he "es.


16/22

A""e" "o !n"ega"e &(x)xnRx

n + 1

gnoe !ncoec" no"a"!on (e.g. !nc%s!on o& !n"ega s!gn) M1

o.e.

Acce"abe a"ena"!es !nc%de

+++++++++ xxx

cxxx

xxx

xxx

'

3

3

4

M'

3

3

4

M'

3

3

4

M'

4

34341

343

4

@.B. & "he cand!da"e has !""en "he !n"ega (e!"hex

xx'

3

3

4

34

++

o ha" "he "h!n !s "he !n"ega) !n a" (a), !" a no" be e!""en

!n (b), b%" "he as a be aaded !& "he !n"ega !s %sed !n (b). A1

#%bs"!"%"!ng 2 and 1 !n"o an &%nc"!on o"he "hanx3+ 3x

2+ '

and s%b"ac"!ng e!"he a o%nd.

#o %s!ng "he! &(x) o &(x)o "he! &(x)dx o "he! &(x)dx! ga!n "he M a (beca%se none o& "hese ! g!e x

3+ 3x

2+ ').

M%s" s%bs"!"%"e &o axs b%" co%d ae a s!.

4 + 8 + 10 41

+ 1 + ' (&o eae) !s acce"abe &o e!dence

o& s%b"ac"!on (S!n!s!be bace"s). M1

o.e. (e.g. 1'4

3

, 1'.7', 4

63

)

M%s" be a s!nge n%be (so 22 64

1

!s A0). A1

Anse on !s M0A0M0A0

aes

cxxx

+++ '4

34

M1A1 4

4x

+x3+ 'x + cM1 A1

4 + 8 + 10 + c (4

1

+ 1 + ' + c) M1 x = 2, 22 + c

= 1'4

3

A1 x = 1, 64

1

+ c M0 A0

(no s%b"ac"!on)

)'31('232d)&( 23

2

1++++= xx M0A0, M0

= 2' 9

= 16 A0

(#%bs"!"%"!ng 2 and 1 !n"ox3+ 3x

2+ ', so 2nd M0)



17/22

[ ]2122

1

63d)66( xxxx +=+M0 A0

[ ]21232

1

2 3d)63( xxxxx +=+M0 A0

= 12 + 12 (3 + 6) M1 A0 = 8 + 12 (1 + 3) M1 A0

4

4x

+x3

+ 'x M1 A1

'14

12'2

4

2 34

34

++++M1

(one nega"!e s!gn !s s%&&!c!en" &o e!dence o& s%b"ac"!on)

= 22 6 4

31'

4

1 =A1

(ao Secoe, !!ng s"%den" as %s!ng S!n!s!be bace"s)

(a) &(x) =x3+ 3x

2+ '

&(x) =xxx '

43

4

++B1M0A0

(b)'1

4

12'2

4

2 34

34

++M1A1M1

= 1'4

3

A1

-he cand!da"e has !""en "he !n"ega !n a" (a). " !s no" e!""en

!n (b), b%" "he as a be aaded as "he !n"ega !s %sed !n (b).[7]

22. y =x(x2 6x + ')

=x3 6x

2+ 'x M1, A1

2

'

3

6

4d)'6(

33423 xxx

xxxx +=+M1, A1&"

4

30

2

'2

4

1

2

'2

4

1

0

23

4

=

+=

+ xxx

M1

411

43)10164(

2'2

4

2

1

2

3

4

=+=

+ xxx

M1, A1(bo"h)

"o"a aea = 411

4

3+

M1

2

7=

o.e. A1cso 9



18/22

A""e" "o %"! o%", %s" be a c%b!c. M1

Aad A a &o "he! &!na es!on o& eans!on (b%" &!na es!on

does no" need "o hae !e "es coec"ed). A1

A""e" "o !n"ega"exnxn+1. Teneo%s a &o soe %se o&

!n"ega"!on, so e.g.

= xx

x

xx

xxxx '222d)')(1(

222

o%d ga!n e"hod a. M1

F" on "he! &!na es!on o& eans!on o!ded !" !s !n "he &o

ax!

' bx*+ ....

n"egand %s" hae a" eas" "o "es and a "es %s" be !n"ega"ed

coec".

& "he !n"ega"e "!ce (e.g.2

1

1

0

and

) and ge" d!&&een" anses, "ae

"he be""e o& "he "o. A1&"

#%bs"!"%"es and s%b"ac"s (e!"he a o%nd) &o one !n"ega.

n"ega %s" be a Schanged &%nc"!on. !"he 1 and 0, 2 and 1 o 2 and 0.

Fo[ ]1

0 0 &o bo""o !!" can be !!ed (o!ded "ha" !" !s 0). M1

M1 #%bs"!"%"es and s%b"ac"s (e!"he a o%nd) &o "o !n"egas.

n"ega %s" be a Schanged &%nc"!on. M%s" hae 1 and 0 and 2 and 1

(o 1 and 2).

-he "o !n"egas do no" need "o be "he sae, b%" "he %s" hae

coe &o a""e"s "o !n"ega"e "he sae &%nc"!on. M1

+=

2

1

23

4

2

1

1

2

2

1

.2

'2

4)&(hee))&(

o)&(o)&(%s!ng(!&o4

11and

4

3o))&(%s!ng(!&o.e.

4

11and

4

3

xx

xxx

xxx

-he anse %s" be cons!s"en" !"h "he !n"ega "he ae %s!ng

(so =

2

1 4

11)&(x

oses "h!s A and "he &!na A).

4

11

a no" be seen e!c!".


19/22

'"h

M1 J "he! a%e &o[ ] [ ]2

1

1

0 &o%e"he! a+

Ueenden" on a" eas" one o& "he a%es co!ng &o !n"ega"!on

(o"he a coe &o e.g. "ae!% %es).

-h!s can be aaded een !& bo"h a%es aead os!"!e. M1

2

7

o.e. @.B. c.s.o. A1 cso[9]

23.

=

2

1d

2

1

2

1x

xx

(G e$%!aen", s%ch as xx 2o,2 2

1

) M1A1

242282

2

1

8

1

2

1

+==

x

/o 42 2, o 2(22 1), o 2(1 + 22) M1A1 4

1s"M1

2

1

2

1

)xx

, 0.

2nd

M #%bs"!"%"!ng !!"s 8 and 1 !n"o a Schanged &%nc"!on

(!.e. no"

2

1

o1

xx ), and s%b"ac"!ng, e!"he a o%nd.

2nd

A -h!s &!na a !s s"! scoed !& 2 + 42 !s eached !a a dec!a.



20/22

@.B. n"ega"!on cons"an" +4a aea, e.g.8

1

2

1

2

1

+

4x

= (28 + 4) (2 + 4) = 2 + 42 (#"! &% as)B%"... a &!na anse s%ch as 2 + 42 + 4!s A0.

@.B. " ! soe"!es be necessa "o S!gnoe s%bse$%en" o!ng

(!s) a&"e a coec" &o !s seen, e.g.

=

2

1d

2

1

2

1x

xx

(M1 A1),

&ooed b !ncoec" s!!&!ca"!on

=

=

2

12

1

2

1

2

1

2

1d x

xxx

(s"! M1 A1).... -he second M a !s s"! aa!abe &o

s%bs"!"%"!ng 8 and 1 !n"o

2

1

2

1x

and s%b"ac"!ng.[4]

24. (a) !"he so!ng 0 =x(6 x) and sho!ngx= 6 (andx=0) B1 1

(b) o sho!ng (6, 0) (andx= 0) sa"!s&!esy = 6xx2

/ao &o sho!ngx= 6

#o!ng 2x = 6xx2(x

2= 4x) "ox= >. M1

x = 4 ( andx= 0) A1


21/22

(c) (Aea =)

)4(

)0(

2 d)6( xxx!!"s no" e$%!ed M1


22/22

-he &!na a a aso be scoed b e!&!ng "ha"0

d

d=

x

y

a"x= 2.

(b) Aea o& "!ange =222

2

1 (M

Documents

Core 2 - Ch 11 - 1 - Integration - Solutions