Cambridge International AS and A Level Mathematics Pure Mathematics 1

P1^

6

Finding

(^) the
(^) area
(^) under
(^) a
(^) curve
Integrating, A = x^6 + 6 x + c.
When x = −1, the line PQ coincides with the left-hand boundary so A = 0
⇒ 0 = 1 − 6 + c
⇒ c = 5.
So A = x^6 + 6 x + 5.
The required area is found by substituting x = 2
A = 64 + 12 + 5
= 81 square units.
Note
The term ‘square units’ is used since area is a square measure and the units are
unknown.
Standardising the procedure
Suppose that you want to find the area between the curve y = f(x), the x axis, and
the lines x = a and x = b. This is shown shaded in figure 6.5.
●●d
d

A

x

= y = f(x).

●●Integrate f(x) to give A = F(x) + c.

●●A = 0 when x = a ⇒ 0 = F(a) + c
⇒ c = −F(a)
⇒ A = F(x) − F(a).

●●The value of A when x = b is F(b) − F(a).

Notation

F(b) − F(a) is written as [(Fx)]ba.

O b x

y

a

y = f(x)

Figure 6.5