Cambridge International AS and A Level Mathematics Pure Mathematics 1

Integration

P1^

6

Notation

This process can be expressed more formally. Suppose you have n rectangles,

each of width δx. Notice that n and δx are related by

nδx = width of required area.

So in the example above,

n δx =  5    −  1    = 4.

In the limit, as n → ∞, δx → 0, the lower

estimate → A and the higher estimate → A.

The area δA of a typical rectangle may be

written yiδx where yi is the appropriate

y value (see figure 6.9).

So for a finite number of strips, n, as shown in figure 6.10, the area A is given

approximately by

A  δA 1     + δA 2  + ... + δAn

or  A  y 1 δx + y 2 δx + ... + ynδx.

This can be written as  A  δAi

i

in

=

=

∑

1

or   A  yxi

i

in

δ.

=

=

∑

1

In the limit, as n → ∞ and δx → 0, the result is no longer an approximation; it is

exact. At this point, A  (^) Σ yi δx is written A = (^) ∫y dx, which you read as ‘the
integral of y with respect to x’. In this case y = x^2 + 1, and you require the area for
values of x from 1 to 5, so you can write
A = (^) ∫ 15 (x^2 + 1)dx.
yi
δx
δAi  yiδx
Figure 6.9
Σ means ‘the sum of’ so
all the δAi are added from
δA 1 (given by i = 1) to δAn
(when i = n).
y
yn
y 4
y 3
y 2
y 1
δA 1 δA 2 δA 3 δA 4
O x
δAn
Figure 6.10