Cambridge International AS and A Level Mathematics Pure Mathematics 1

Integration

P1^

6

EXAMPLE 6.5 Find the area between the curve y =     20   −  3 x^2 , the x axis and the lines x =    1    and x = 2.

SOLUTION

f(x) =  20   −  3 x^2 ⇒ F(x) =  20 x − x^3

a =     1    and b =    2

⇒   Area    = [–^20 xx^3 ] 12

= (40 − 8) − (20 − 1)

=   13   square units.

Area as the limit of a sum

Suppose you want to find the area between the curve y = x^2 + 1, the x axis and the

lines x =   1    and x = 5. This area is shaded in figure 6.6.

You can find an estimate of the shaded area, A, by considering the area of four

rectangles of equal width, as shown in figure 6.7.

A

x

y

y = x^2 + 1

O 1 5

Figure 6.6

x

y y = x (^2) + 1
0
2
5
10
17
1 2 3 4 5
Figure 6.7