Cambridge Additional Mathematics

Integration (Chapter 15) 433

7aUse the identity cos^2 x=^12 +^12 cos(2x) to help evaluate

Z¼

4

0

cos^2 xdx.

b Use the identity sin^2 x=^12 ¡^12 cos(2x) to help evaluate

Z ¼

2

0

sin^2 xdx.

8 Evaluate the following integrals using area interpretation:

a

Z 3

0

f(x)dx b

Z 7

3

f(x)dx

c

Z 4

2

f(x)dx d

Z 7

0

f(x)dx

9 Evaluate the following integrals using area interpretation:

a

Z 4

0

f(x)dx b

Z 6

4

f(x)dx

c

Z 8

6

f(x)dx d

Z 8

0

f(x)dx

10 Write as a single integral:

a

Z 4

2

f(x)dx+

Z 7

4

f(x)dx b

Z 3

1

g(x)dx+

Z 8

3

g(x)dx+

Z 9

8

g(x)dx

11 a If

Z 3

1

f(x)dx=2 and

Z 6

1

f(x)dx=¡ 3 , find

Z 6

3

f(x)dx.

b If

Z 2

0

f(x)dx=5,

Z 6

4

f(x)dx=¡ 2 , and

Z 6

0

f(x)dx=7, find

Z 4

2

f(x)dx.

12 Given that

Z 1

¡ 1

f(x)dx=¡ 4 , determine the value of:

a

Z¡ 1

1

f(x)dx b

Z 1

¡ 1

(2 +f(x))dx c

Z 1

¡ 1

2 f(x)dx

d ksuch that

Z 1

¡ 1

kf(x)dx=7

13 If g(2) = 4 and g(3) = 5, calculate

Z 3

2

(g^0 (x)¡1)dx.

2

-2

2

4

y

6 x

y = f(x)

O

2

-2

2

4

y

68 x

y = f(x)

O

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_15\433CamAdd_15.cdr Monday, 7 April 2014 4:00:02 PM BRIAN