Cambridge International AS and A Level Mathematics Pure Mathematics 1

Integration

P1^

6

!   Since the integration is ‘with respect to x’, indicated by the dx and the fact that

the limits a and b are values of x, it cannot be evaluated unless the function y is

also written in terms of x.

EXAMPLE 6.18 The region between the curve y = x^2 , the x axis and the lines x =    1    and x =    3    is

rotated through 360° about the x axis.

Find the volume of revolution which is formed.

SOLUTION

The region is shaded in figure 6.29.

Using V = (^) ∫
a
b
πy^2 dx
volume = (^) ∫
1
3
π(x^2 )^2 dx
= (^) ∫
1
3
πx^4 dx
= πx
5
1
3
5









= π 5 (– 243 1 )

=   2425 π.

The volume is   2425 π cubic units or   152  cubic units (3 s.f.).

!   Unless a decimal answer is required, it is usual to leave π in the answer, which is

then exact.

O 1 3

y

x

y = x^2

Figure 6.29

Since in this case

y = x^2

y^2 = (x^2 )^2 = x^4.