Cambridge Additional Mathematics

Applications of integration (Chapter 16) 441

Example 4 Self Tutor

Use

Zb

a

[yU¡yL]dx to find the area bounded by thex-axis and y=x^2 ¡ 2 x.

The curve cuts thex-axis when y=0

) x^2 ¡ 2 x=0

) x(x¡2) = 0

) x=0or 2

) thex-intercepts are 0 and 2.

Area=

Z 2

0

[yU¡yL]dx

=

Z 2

0

[0¡(x^2 ¡ 2 x)]dx

=

Z 2

0

(2x¡x^2 )dx

=

·

x^2 ¡

x^3

3

̧ 2

0

=

¡

4 ¡^83

¢

¡(0)

) the area is^43 units^2.

Example 5 Self Tutor

Find the area of the region enclosed by y=x+2and y=x^2 +x¡ 2.

y=x+2meets y=x^2 +x¡ 2

where x^2 +x¡2=x+2

) x^2 ¡4=0

) (x+ 2)(x¡2) = 0

) x=§ 2

Area=

Z 2

¡ 2

[yU¡yL]dx

=

Z 2

¡ 2

[(x+2)¡(x^2 +x¡2)]dx

=

Z 2

¡ 2

(4¡x^2 )dx

=

·

4 x¡

x^3

3

̧ 2

¡ 2

=

¡

8 ¡^83

¢

¡

¡

¡8+^83

¢

=10^23 units^2

) the area is 1023 units^2 :

2

y

O x

y=0U

y=x-2xL 2

-2^12

2

-2

y

O x

y=x+2

y=x +x-2 2

4037 Cambridge

cyan magenta yellow black Additional Mathematics

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\CAM4037\CamAdd_16\441CamAdd_16.cdr Monday, 7 April 2014 4:17:36 PM BRIAN