Finding(^) volumes
(^) by
(^) integration
209
P1^
6
●?^1 Describe^ the^ solid^ of^ revolution^ obtained^ by^ a^ rotation^ through^ 360°^ of
(i) a rectangle about one side
(ii) a semi-circle about its diameter
(iii) a circle about a line outside the circle.
●^2 Calculate^ the^ volume^ of^ the^ solid^ obtained^ in^ figure^ 6.26,^ leaving^ your^ answer^
as a multiple of π.
Solids formed by rotation about the x axis
Now look at the solid formed by rotating the shaded region in figure 6.27
through 360° about the x axis.
The volume of the solid of revolution (which is usually called the volume of
revolution) can be found by imagining that the solid can be sliced into thin discs.
The disc shown in figure 6.28 is approximately cylindrical with radius y and
thickness δx, so its volume is given by
δV = πy^2 δx.
The volume of the solid is the limit of the sum of all these elementary discs as
δx → 0,
i.e. the limit as δx → 0 of
overall
discs
∑ δV
=
xaxb==
∑πy^2 δx.The limiting values of sums such as these are
integrals soV = (^) ∫
b
aπy
(^2) dx
The limits are a and b because x takes values from a to b.
O a x
y y = f(x)
b
Figure 6.27
O x
y
Figure 6.28
You can write this as
V = ∫x=b x=a πy^2 dx
emphasising that the limits
a and b are values of x, not y.