Cambridge International AS and A Level Mathematics Pure Mathematics 1

Finding

(^) volumes
(^) by
(^) integration
211
P1^
6
Solids formed by rotation about the y axis
When a region is rotated about the y axis a very different solid is obtained.
Notice the difference between the solid obtained in figure 6.31 and that in
figure 6.28.
For rotation about the x axis you obtained the formula
Vx axis = (^) ∫
b
aπy
(^2) dx.
In a similar way, the formula for rotation about the y axis
Vy axis = (^) ∫
q
pπx
(^2) dy can be obtained.
In this case you will need to substitute for x^2 in terms of y.
●^ How^ would^ you^ prove^ this^ result?
EXAMPLE 6.19 The region between the curve y = x^2 , the y axis and the lines y = 2 and y = 5 is
rotated through 360° about the y axis.
Find the volume of revolution which is formed.
SOLUTION
The region is shaded in figure 6.32.
Using V = (^) ∫
q
pπx
(^2) dy
volume = (^) ∫^52 πy dy since x^2 = y
(^) =






πy^2

2

5

2

= π 2    (25 − 4)

=^21

π cubic units.

O x

y y = f(x)

p

q

Figure 6.30

O x

y

Figure 6.31

O

y

x

y = x^2

2

5

Figure 6.32