Cambridge Additional Mathematics

438 Applications of integration (Chapter 16)

Opening problem

#endboxedheading

The illustrated curves are those of y= sinx and

y= 3 sinx.

Things to think about:

a Can you identify each curve?

b Can you find the shaded area enclosed by C 1 and C 2

for 06 x 6 ¼?

We have already seen how definite integrals can be related to the areas between functions and thex-axis. In

this chapter we explore this relationship further, and consider other applications of integral calculus including

kinematics.

We have already established inChapter 15that:

If f(x) is positive and continuous on the interval

a 6 x 6 b, then the area bounded by y=f(x), the

x-axis, and the vertical lines x=a and x=b is

given by A=

Zb

a

f(x)dx or

Zb

a

ydx.

Example 1 Self Tutor

Find the area of the region enclosed by y=2x, thex-axis, x=0, and x=4by using:

a a geometric argument b integration.

a

Area=^12 £ 4 £ 8

=16units^2

b Area=

Z 4

0

2 xdx

=

£

x^2

¤ 4

0

=4^2 ¡ 02

=16units^2

EXERCISE 16A

1 Find the area of each of the regions described below by using:

i a geometric argument ii integration

a y=5, thex-axis, x=¡ 6 , and x=0

b y=x, thex-axis, x=4, and x=5

c y=¡ 3 x, thex-axis, x=¡ 3 , and x=0

d y=¡x, thex-axis, x=0, and x=2

A THE AREA UNDER A CURVE

y

O abx

y = f(x)

A

y

4 x

8

O

y=2x

y

x

C 2

C 1

O ¼

cyan magenta yellow black

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_16\438CamAdd_16.cdr Monday, 7 April 2014 4:17:15 PM BRIAN