Cambridge International AS and A Level Mathematics Pure Mathematics 1

P1^

6

The area between two curves

The area between two curves

EXAMPLE 6.13  Find  the area    enclosed    by  the line    y = x + 1   and the curve   y = x^2 −   2 x +   1.

SOLUTION

First   draw    a   sketch  showing where   these   graphs  intersect   (see    figure  6.18).

When    they    intersect

x^2 −   2 x +   1    = x +  1

⇒ x^2 −     3 x =   0

⇒ x (x −    3)  =   0

⇒ x =   0   or  x   =   3.

The shaded  area    can now be  found   in  one of  two ways.

Method 1

Area    A   can be  treated as  the difference  between the two areas,  B   and C,  shown   in      

figure  6.19.

y

O 1 3 x

A

y = x^2 – 2x + 1

y = x + 1

Figure 6.18

B

y

O 1 3 x

C

y

O 1 3 x

y = x + 1 y = x^2 – 2x + 1

Figure 6.19