Cambridge International AS and A Level Mathematics Pure Mathematics 1

Integration

P1^

6

EXAMPLE 6.11 Find the area of the region bounded by the curve with equation y

x

=−^223 , the

lines x =   2    and x = 4, and the x axis.

SOLUTION

The region in question is shaded in figure 6.16.

The shaded area is

A

x

x

xx

x x

=−()

=−()

=

−









∫

∫

−

−

2

4

2

2

(^42)
1
2
(^23)

23

2

1

3

d

d

()

–

44

2

2 4

3

12 16

55

1

2

=−







=−()− −− −

=−

x

• x

()

.

Therefore the shaded area is 5.5 square units, and it is below the x axis.

y

x

2 4

y = – 3x^22

O

Figure 6.16