4

2nd Year HL Math

QUIZ Integration

27/09/10

42 min

25 marks

Name:_____________________________

Find

x

x

d

n

1

.

Working:

Answers:

..

(Total 3 marks)

2.Find the real number k > 1 for which

+

k

x

1

2

1

1

dx =

2

3

.

Working:

Answers:

..............

(Total 4 marks)

3.The area of the enclosed region shown in the diagram is defined by

y x2 + 2, y ax + 2, where a > 0.

x

y

a

0

2

This region is rotated 360 about the x-axis to form a solid of revolution. Find, in terms of a, the volume of this solid of revolution.

Working:

Answers:

..

(Total 4 marks)

4.Using the substitution u =

2

1

x + 1, or otherwise, find the integral

+

1

2

1

x

x

dx.

Working:

Answers:

..

(Total 4 marks)

Please, answer this question on a separate sheet of paper.

5.A uniform rod of length l metres is placed with its ends on two supports A, B at the same horizontal level.

x

metres

l

metres

yx

()

A

B

If y (x) metres is the amount of sag (ie the distance below [AB]) at a distance x metres from support A, then it is known that

(

)

lx

x

l

x

y

125

1

d

d

2

3

2

2

=

.

(a)(i)Let z =

1500

1

2

3

125

1

2

3

3

+

-

lx

x

l

. Show that

x

z

d

d

=

(

)

lx

x

l

-

2

3

125

1

.

Given that

x

w

d

d

= z and w(0) = 0, find w(x).

(iii)Show that w satisfies

2

2

d

d

x

w

=

3

125

1

l

(x2 lx), and that w(l) = w(0) = 0.

(8)

(b)Find the sag at the centre of a rod of length 2.4 metres.

(2)

(Total 10 marks)