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INTEGRATION

PAPER 1

1. Evaluate

2 3

2 2

1 2

( 4 ) ( 4 )x x dx x x dx

+ . [3

marks]

Answer: ...

2. Given that 23 .dy

tdt = Expressy in terms of t if 2=t and6=y . [3

marks]

Answer: ...

3. Given that =5

1

8dx)(xg , find

(a) the value of 1

5

dx)(xg ,

(b) the value of k if =5

1

10dx)]([ xgkx . [4 marks]

Answer: ...

1

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4. Find

3

1

( 4)x x dx

+ . [3 marks]

Answer: ...

5. Given that

2

1

( ) 5f x dx

= . Find the value ofkif2

1

1[ ( ( ) ]

2

k f x x dx

= . [3 marks]

Answer: ...

6. Find

dxx

x3

54

. [2 marks]

Answer: ...

2

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7. Diagram 1 shows part of the graph of the curve24xy = and the straight line px = . Find

the value of p if the area of the shaded region is 36 unit2. [3 marks]

DIAGRAM 1

Answer: ...

8. Given that ++=+cxkdx

x

n)21()21(

85 . Find the value ofkand n. [3 marks]

Answer: k= ...

n = ...

3

y

0x

y = 4x2

x = p

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9. Evaluate

2

2

1

(3 1)x dx

. [2 marks]

Answer: ...

10. Diagram 2 shows the curve ( )y f x= cutting thex-axis at andx a x b= = .

DIAGRAM 2

Given that the area of the shaded region is 5 unit2, find the value of b

a

xf dx)(2 .

[2 marks]

Answer: .

4

y

O xa b

( )y f x=

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PAPER 2

11. (a) Diagram 3 shows part of the curve3)2(27 = xy intersecting the x-axis at point

P. Find,

(i) coordinates of P, [2 marks]

(ii) the area of the shaded region. [3 marks]

DIAGRAM 3

(b) Diagram 4 shows the shaded region bounded by the curve 43 += xy , the linex = k,

and x = 4. When the shaded region is rotated through 360o about the x-axis, the

volume generated is 26

unit

3

. Find the value ofk. [5 marks]

DIAGRAM 4

5

3)2(27 = xy

y

O xP

y

0 k 4 x

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DIAGRAM 5

12. (a) Diagram 5 shows the curve xy =2

, the line 3=y and 2=y . Find the area of the

shaded region. [4 marks]

DIAGRAM 6

(b) In Diagram 6, the curve2

4 xxy = has a maximum point P. Find the volume

generated when the shaded region is rotated through 4 right angles about thex-axis.

[6 marks]

6

y

x

y2 =x

y = 3

y = - 2

0

y = 4x x2

P

y

x0

y

x0 A

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DIAGRAM 7

13. (a) Diagram 7 shows part of the curve kxxy += 123 where A is the minimum point.

Find

(i) the value of k, [2 marks]

(ii) the area of the shaded region. [3 marks]

DIAGRAM 8

(b) Diagram 8 shows part of the curve 267 xxy += which intersects the straight lineky = at the point (6, k). Calculate the area of the shaded region. [5 marks]

7

y = k

267 xxy +=

(6, k)

y

x0

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14. Diagram 9 shows the straight line 4y x= + intersecting the curve 2( 2)y x= at the points

A andB.

DIAGRAM 9

Find(a) the value of k, [2 marks]

(b) the area of the shaded regionP, [5 marks]

(c) the volume generated, in terms of , when the shaded region Q is revolved 360about thex-axis. [3 marks]

8

y

xkO

A

B

4y x= +

2( 2)y x=

P

Q

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Answer

1.3

26

2. 23 = ty

3. (a) 8

(b)2

3

4.

3

125

5.5

2=k

6. cx

x+

3

23

2

7.3=p

8. 1=k and 4=n

9. 6

10. 10

11. (a) (i) (5, 0)

(ii)2

4

475unit

(b) 2=k

12. (a)2

3

211 unit

(b)

15

256unit2

13. (a) (i) k=16

(ii) 12 unit2

(b) 36 unit2

14. (a) 5

(b) 20.83

(c) 532

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