Cambridge International AS and A Level Mathematics Pure Mathematics 1

Integration

P1^

6

A = B − C

= (^) ∫
3
0 (x^ +^ 1)^ dx^ −^ ∫
3
0 (x
(^2) − 2 x + 1) dx
(^) =+






 +







=+() 

x^2 x x xx

0

(^33)
2
0
3
23
92 30 27

––

–– 33

9

2

()–– 93 + 0 

= squareunits.

Method 2

(^) A x
xx x

=

=+ +

∫{ }

()–( –)

0 topcurve–bottomcurve d

3

((^1221 )

=

=







= 

∫

∫

d

d

x

xx x

xx

0

3

2

0

3

23

0

3

3

3

23

(^2729)

(– )

–

– 

=

–[]

.

0

9

2 squareunits

EXERCISE 6D  1  The diagram shows the curve

y = x^2 and the line y = 9.

The enclosed region has been shaded.

(i) Find the two points of

intersection (labelled A and B).

(ii) Using integration, show that

the area of the shaded region

is  36   square units.

y

O 1 3 x

y = x^2 – 2x + 1

y = x + 1

Figure 6.20

The height of this rectangle

is the height of the top

curve minus the height of

the bottom curve.

y

O x

A B

y = x2

y = 9