Cambridge Additional Mathematics

Applications of integration (Chapter 16) 439

Example 2 Self Tutor

Find the area of the region enclosed by y=x^2 +1, thex-axis,x=1,

and x=2.

Area=

Z 2

1

(x^2 +1)dx

=

·

x^3

3

+x

̧ 2

1

=

¡ 8

3 +2

¢

¡

¡ 1

3 +1

¢

=3^13 units^2

2 Find the area of the region bounded by:

a y=x^2 , thex-axis, and x=1

b y= sinx, thex-axis, x=0, and x=¼

c y=x^3 , thex-axis, x=1, and x=4

d y=ex, thex-axis, they-axis, and x=1

e thex-axis and the part of y=6+x¡x^2 above thex-axis

f the axes and y=

p

9 ¡x

g y=

1

x^2

, thex-axis, x=1, and x=2

h y=2¡

1

p

x

, thex-axis, and x=4

i y=ex+e¡x, thex-axis, x=¡ 1 , and x=1

Example 3 Self Tutor

Find the area enclosed by one arch of the curve y= sin 2x and thex-axis.

The period of y= sin 2x is^22 ¼=¼, so the first positivex-intercept is¼ 2.

The required area=

Z ¼

2

0

sin 2xdx

=

h

1

2 (¡cos 2x)

i¼ 2

0

=¡^12

h

cos 2x

i¼ 2

0

=¡^12 (cos¼¡cos 0)

=1unit^2

3 Find the area enclosed by one arch of the curve y= cos 3x and thex-axis.

GRAPHING

PACKAGE

It is helpful to

sketch the region.

Use the graphing package

to check your answers.

1 2

y

O x

y=x +1 2

¼ x

y

O w_p

y=sin2x

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_16\439CamAdd_16.cdr Monday, 7 April 2014 4:19:08 PM BRIAN