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