Cambridge International AS and A Level Mathematics Pure Mathematics 1

Integration

P1^

6

If  T   is  close   to  P it    is  appropriate to  use the notation    δx  (a  small   change  in  x)

for the difference  in  their   x   co-ordinates    and δy  for the difference  in  their   y

co-ordinates.   The area    shaded  in  figure  6.3 is  then    referred    to  as  δA  (a  small   

change  in  A).

This    area    δA  will    lie between the areas   of  the rectangles  PQRS    and UQRT

yδx  δA   (y + δy)δx.

Dividing    by  δx

y  δ

δ

A

x

 y + δy.

In  the limit   as  δx →    0,  δy  also    approaches  zero    so  δA  is  sandwiched  between y   and 

something   which   tends   to  y.

But                 lim             δ

δ

A

x

A

= x

d

d.

δx → 0

This    gives   ddAx^ = y.

Note

This important result is known as the fundamental theorem of calculus: the rate of

change of the area under a curve is equal to the length of the moving boundary.

EXAMPLE 6.4  Find   the area    under   the curve   y =     6 x^5 + 6   between x = −1  and x = 2.

SOLUTION

Let A   be  the shaded  area    which   is  bounded by  the curve,  the x   axis,   and the 

moving  boundary    PQ  (see    figure  6.4).

Then     ddAx^ = y =    6 x^5 + 6.

–1 O 2 x

y

Q

6

P

(xy)

Figure 6.4 

Notice that the

curve crosses

the x axis when

x = –1.