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Section 5.2: Volumes of Solids of Revolution

In the previous sections, we used Riemann sums to approximate the area under a curve and

then deﬁned a deﬁnite integral as the exact area. We will use the same basic technique in

this section to ﬁnd the volume of certain solid shapes, called solids of revolution.

Def: Asolid of revolution is formed by revolving a planar region around a line in the

plane called its axis of revolution.

Method 1: Disks

When y=f(x)is a continuous function on [a, b]and Ais the planar region under the graph

of fand above the x-axis from x=ato x=b(area under the curve) and when we revolve

this region around the x-axis, we will use the disk method.

To ﬁnd the volume, we slice the solid into nvertical pieces, intersecting the x-axis at xi=

a+i△xwhere △x=b−a

n. We approximate the volume of the slice using a cylinder of

height △x.

1

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The volume of a cylinder is V=πr2h. Then the volume of this shape, using ndisks

(cylinders) to approximate, is

V≈

n

X

i=1

π(f(xi))2△x.

As with our deﬁnition of area, the exact volume of this shape is obtained by taking the limit

as nincreases without bound.

V= lim

n→∞

n

X

i=1

π(f(xi))2△x=Zb

a

π(f(x))2dx

To aid memory, we can label r(x) = f(x), where ris radius. Then

V=Zb

a

π(r(x))2dx.

Ex: Find the volume of the solid of revolution generated by revolving the region bounded

by f(x) = √x,x= 1,x= 4 and the x-axis, about the x-axis.

Let’s look at the graph and the solid it generates:

( )=√= =

It should be easy enough to see that a vertical slice of this solid is roughly cylindrical, so the

disk method will work.

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

V=Zb

a

π(r(x))2dx

=Z4

1

π(√x)2dx

=Z4

1

πx dx

=π·1

2x2

4

1

=π

2(42−12)

=π

2(15)

=15π

2

≈23.6

Since the units are not speciﬁed here, we can simply say 23.6 cubic units of volume.

Notice that we could also revolve a function x=g(y)about the y-axis to generate a solid of

revolution.

Here, the volume is given by

V=Zd

c

π(g(y))2dy =Zd

c

π(r(y))2dy.

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