Cambridge International AS and A Level Mathematics Pure Mathematics 1

P1^

6

Area as the limit of a sum

ACTIVITY 6.2  Figure 6.13 shows the region bounded by the graph of y = x + 3, the x axis and the
lines x = a and x = b.

(i) Find    the shaded  area,   A,  by  considering it  as  the difference  between the two 

trapezia    shown   in  figure  6.14.

(ii)    Show    that    the expression  for A   you obtained    in  part    (i) may be  written as  

x x

a

2 b

2

 + 3







.

(iii) Show  that    you obtain  the same    answer  for A   by  integration.

O a b x

3

y

y = x + 3

Figure 6.13

O a b x

3

b + 3

y

y = x + 3

O a x

3

y

y = x + 3

a + 3

Figure 6.14