INTEGRATIONPAPER 1

1. Evaluate . [3

marks]

Answer : …...………………………

2. Given that Express y in terms of t if and . [3

marks]

Answer : …...………………………

3. Given that , find

(a) the value of ,

(b) the value of k if . [4 marks]

Answer : …...………………………

1

4. Find . [3 marks]

Answer: …...………………………

5. Given that . Find the value of k if . [3 marks]

Answer: …...………………………

6. Find . [2 marks]

Answer: …...………………………

2

7. Diagram 1 shows part of the graph of the curve and the straight line . Find the value of p if the area of the shaded region is 36 unit2. [3 marks]

DIAGRAM 1

Answer: …...………………………

8. Given that . Find the value of k and n. [3 marks]

Answer: k = ..…………………….

n = …..………………….

3

y

0 x

y = 4x2

x = p

9. Evaluate . [2 marks]

Answer: …...………………………

10. Diagram 2 shows the curve cutting the x-axis at .

DIAGRAM 2

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

[2 marks]

Answer: …….………………………

4

y

O xa b

( )y f x

PAPER 2

11. (a) Diagram 3 shows part of the curve 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 , the line x = k,

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

volume generated is 26 unit3. Find the value of k. [5 marks]

DIAGRAM 4

5

3)2(27 xy

y

O xP

y

0 k 4 x

DIAGRAM 5

12. (a) Diagram 5 shows the curve , the line and . Find the area of the

shaded region. [4 marks]

DIAGRAM 6

(b) In Diagram 6, the curve has a maximum point P. Find the volume

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

[6 marks]

6

y

x

y2 = x

y = 3

y = - 2

0

y = 4x – x2

P y

x 0

y

x0 A

DIAGRAM 7

13. (a) Diagram 7 shows part of the curve 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 which intersects the straight line at the point (6, k). Calculate the area of the shaded region. [5 marks]

7

y = k

267 xxy

(6, k)

y

x0

14. Diagram 9 shows the straight line intersecting the curve at the points A and B.

DIAGRAM 9Find(a) the value of k, [2 marks](b) the area of the shaded region P, [5 marks](c) the volume generated, in terms of π , when the shaded region Q is revolved 360˚

about the x-axis. [3 marks]

8

y

xkO

A

B4y x

2( 2)y x

PQ

Answer

9

1.

2.

3. (a) 8

(b)

4.

5.

6.

7.

8. and

9. 6

10. 10

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

(ii)

(b)

12. (a)

(b) unit2

13. (a) (i) k =16

(ii) 12 unit2

(b) 36 unit2

14. (a) 5

(b) 20.83

(c)

1

2