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In this section, we will learn:

How to apply the method of cylindrical shells to find out the volume of a solid.

How to apply the method of cylindrical shells to find out the volume of a solid.

1.
6

APPLICATIONS OF INTEGRATION

APPLICATIONS OF INTEGRATION

2.
APPLICATIONS OF INTEGRATION

6.3

Volumes by

Cylindrical Shells

In this section, we will learn:

How to apply the method of cylindrical shells

to find out the volume of a solid.

6.3

Volumes by

Cylindrical Shells

In this section, we will learn:

How to apply the method of cylindrical shells

to find out the volume of a solid.

3.
VOLUMES BY CYLINDRICAL SHELLS

Some volume problems are very

difficult to handle by the methods

discussed in Section 6.2

Some volume problems are very

difficult to handle by the methods

discussed in Section 6.2

4.
VOLUMES BY CYLINDRICAL SHELLS

Let’s consider the problem of finding the

volume of the solid obtained by rotating about

the y-axis the region bounded by y = 2x2 - x3

and y = 0.

Let’s consider the problem of finding the

volume of the solid obtained by rotating about

the y-axis the region bounded by y = 2x2 - x3

and y = 0.

5.
VOLUMES BY CYLINDRICAL SHELLS

If we slice perpendicular to the y-axis,

we get a washer.

However, to compute the inner radius and the outer

radius of the washer,

we would have to

solve the cubic

equation y = 2x2 - x3

for x in terms of y.

That’s not easy.

If we slice perpendicular to the y-axis,

we get a washer.

However, to compute the inner radius and the outer

radius of the washer,

we would have to

solve the cubic

equation y = 2x2 - x3

for x in terms of y.

That’s not easy.

6.
VOLUMES BY CYLINDRICAL SHELLS

Fortunately, there is a method—the

method of cylindrical shells—that is

easier to use in such a case.

Fortunately, there is a method—the

method of cylindrical shells—that is

easier to use in such a case.

7.
CYLINDRICAL SHELLS METHOD

The figure shows a cylindrical shell

with inner radius r1, outer radius r2,

and height h.

The figure shows a cylindrical shell

with inner radius r1, outer radius r2,

and height h.

8.
CYLINDRICAL SHELLS METHOD

Its volume V is calculated by subtracting

the volume V1 of the inner cylinder from

the volume of the outer cylinder V2 .

Its volume V is calculated by subtracting

the volume V1 of the inner cylinder from

the volume of the outer cylinder V2 .

9.
CYLINDRICAL SHELLS METHOD

Thus, we have:

V V2 V1

2 2

r2 h r h 1

2 2

(r2 r )h 1

(r2 r1 )(r2 r1 )h

r2 r1

2 h(r2 r1 )

2

Thus, we have:

V V2 V1

2 2

r2 h r h 1

2 2

(r2 r )h 1

(r2 r1 )(r2 r1 )h

r2 r1

2 h(r2 r1 )

2

10.
CYLINDRICAL SHELLS METHOD Formula 1

Let ∆r = r2 – r1 (thickness of the shell) and

r 12 r2 r1

(average radius of the shell).

Then, this formula for the volume of a

cylindrical shell becomes:

V 2 rhr

Let ∆r = r2 – r1 (thickness of the shell) and

r 12 r2 r1

(average radius of the shell).

Then, this formula for the volume of a

cylindrical shell becomes:

V 2 rhr

11.
CYLINDRICAL SHELLS METHOD

V 2 rhr

The equation can be remembered as:

V = [circumference] [height] [thickness]

V 2 rhr

The equation can be remembered as:

V = [circumference] [height] [thickness]

12.
CYLINDRICAL SHELLS METHOD

Now, let S be the solid

obtained by rotating

about the y-axis the

region bounded by

y = f(x) [where f(x) ≥ 0],

y = 0, x = a and x = b,

where b > a ≥ 0.

Now, let S be the solid

obtained by rotating

about the y-axis the

region bounded by

y = f(x) [where f(x) ≥ 0],

y = 0, x = a and x = b,

where b > a ≥ 0.

13.
CYLINDRICAL SHELLS METHOD

Divide the interval [a, b] into n subintervals

[xi - 1, xi ] of equal width xi and let be x

the midpoint of the i th subinterval.

Divide the interval [a, b] into n subintervals

[xi - 1, xi ] of equal width xi and let be x

the midpoint of the i th subinterval.

14.
CYLINDRICAL SHELLS METHOD

The rectangle with

base [xi - 1, xi ] and

height f ( xi ) is rotated

about the y-axis.

The result is a

cylindrical shell with

average radius xi ,

height f ( xi ) , and

thickness ∆x.

The rectangle with

base [xi - 1, xi ] and

height f ( xi ) is rotated

about the y-axis.

The result is a

cylindrical shell with

average radius xi ,

height f ( xi ) , and

thickness ∆x.

15.
CYLINDRICAL SHELLS METHOD

Thus, by Formula 1, its volume is

calculated as follows:

Vi (2 xi )[ f ( xi )]x

Thus, by Formula 1, its volume is

calculated as follows:

Vi (2 xi )[ f ( xi )]x

16.
CYLINDRICAL SHELLS METHOD

So, an approximation to the volume V of S

is given by the sum of the volumes of

these shells:

n n

V Vi 2 xi f ( xi )x

i 1 i 1

So, an approximation to the volume V of S

is given by the sum of the volumes of

these shells:

n n

V Vi 2 xi f ( xi )x

i 1 i 1

17.
CYLINDRICAL SHELLS METHOD

The approximation appears to become better

as n →∞.

However, from the definition of an integral,

we know that:

n b

lim 2 xi f ( xi )x 2 x f ( x)dx

n a

i 1

The approximation appears to become better

as n →∞.

However, from the definition of an integral,

we know that:

n b

lim 2 xi f ( xi )x 2 x f ( x)dx

n a

i 1

18.
CYLINDRICAL SHELLS METHOD Formula 2

Thus, the following appears plausible.

The volume of the solid obtained by rotating

about the y-axis the region under the curve

y = f(x) from a to b, is:

b

V 2 xf ( x)dx

a

where 0 ≤ a < b

Thus, the following appears plausible.

The volume of the solid obtained by rotating

about the y-axis the region under the curve

y = f(x) from a to b, is:

b

V 2 xf ( x)dx

a

where 0 ≤ a < b

19.
CYLINDRICAL SHELLS METHOD

The argument using cylindrical shells

makes Formula 2 seem reasonable,

but later we will be able to prove it.

The argument using cylindrical shells

makes Formula 2 seem reasonable,

but later we will be able to prove it.

20.
CYLINDRICAL SHELLS METHOD

Here’s the best way to remember

the formula.

Think of a typical shell,

cut and flattened,

with radius x,

circumference 2πx,

height f(x), and

thickness ∆x or dx:

b

2 x f( x)

a dx

circumference height thickness

Here’s the best way to remember

the formula.

Think of a typical shell,

cut and flattened,

with radius x,

circumference 2πx,

height f(x), and

thickness ∆x or dx:

b

2 x f( x)

a dx

circumference height thickness

21.
CYLINDRICAL SHELLS METHOD

This type of reasoning will be helpful

in other situations—such as when we

rotate about lines other than the y-axis.

This type of reasoning will be helpful

in other situations—such as when we

rotate about lines other than the y-axis.

22.
CYLINDRICAL SHELLS METHOD Example 1

Find the volume of the solid obtained by

rotating about the y-axis the region

bounded by y = 2x2 - x3 and y = 0.

Find the volume of the solid obtained by

rotating about the y-axis the region

bounded by y = 2x2 - x3 and y = 0.

23.
CYLINDRICAL SHELLS METHOD Example 1

We see that a typical shell has

radius x, circumference 2πx, and

height f(x) = 2x2 - x3.

We see that a typical shell has

radius x, circumference 2πx, and

height f(x) = 2x2 - x3.

24.
CYLINDRICAL SHELLS METHOD Example 1

So, by the shell method,

the volume is: 2

V 2 x 2 x x dx

2 3

0

2

3 4

2 x (2 x x )dx

0

4 5 2

2 x x

1

2

1

5 0

2 8 325 165

So, by the shell method,

the volume is: 2

V 2 x 2 x x dx

2 3

0

2

3 4

2 x (2 x x )dx

0

4 5 2

2 x x

1

2

1

5 0

2 8 325 165

25.
CYLINDRICAL SHELLS METHOD Example 1

It can be verified that the shell method

gives the same answer as slicing.

The figure shows

a computer-generated

picture of the solid

whose volume we

computed in the

example.

It can be verified that the shell method

gives the same answer as slicing.

The figure shows

a computer-generated

picture of the solid

whose volume we

computed in the

example.

26.
Comparing the solution of Example 1 with

the remarks at the beginning of the section,

we see that the cylindrical shells method

is much easier than the washer method

for the problem.

We did not have to find the coordinates of the local

maximum.

We did not have to solve the equation of the curve

for x in terms of y.

the remarks at the beginning of the section,

we see that the cylindrical shells method

is much easier than the washer method

for the problem.

We did not have to find the coordinates of the local

maximum.

We did not have to solve the equation of the curve

for x in terms of y.

27.
However, in other examples,

the methods learned in Section 6.2

may be easier.

the methods learned in Section 6.2

may be easier.

28.
CYLINDRICAL SHELLS METHOD Example 2

Find the volume of the solid obtained

by rotating about the y-axis the region

between y = x and y = x2.

Find the volume of the solid obtained

by rotating about the y-axis the region

between y = x and y = x2.

29.
CYLINDRICAL SHELLS METHOD Example 2

The region and a typical shell

are shown here.

We see that the shell has radius x, circumference 2πx,

and height x - x2.

The region and a typical shell

are shown here.

We see that the shell has radius x, circumference 2πx,

and height x - x2.

30.
CYLINDRICAL SHELLS METHOD Example 2

Thus, the volume of the solid is:

1

V 2 x x x 2 dx

0

1

2 x x dx

2 3

0

3 4 1

x x

2

3 4 0 6

Thus, the volume of the solid is:

1

V 2 x x x 2 dx

0

1

2 x x dx

2 3

0

3 4 1

x x

2

3 4 0 6

31.
CYLINDRICAL SHELLS METHOD

As the following example shows,

the shell method works just as well

if we rotate about the x-axis.

We simply have to draw a diagram to identify

the radius and height of a shell.

As the following example shows,

the shell method works just as well

if we rotate about the x-axis.

We simply have to draw a diagram to identify

the radius and height of a shell.

32.
CYLINDRICAL SHELLS METHOD Example 3

Use cylindrical shells to find the volume of

the solid obtained by rotating about the x-axis

the region under the curve y x from 0 to 1.

This problem was solved using disks in Example 2

in Section 6.2

Use cylindrical shells to find the volume of

the solid obtained by rotating about the x-axis

the region under the curve y x from 0 to 1.

This problem was solved using disks in Example 2

in Section 6.2

33.
CYLINDRICAL SHELLS METHOD Example 3

To use shells, we relabel the curve

y x

as x = y2.

For rotation about

the x-axis, we see that

a typical shell has

radius y, circumference

2πy, and height 1 - y2.

To use shells, we relabel the curve

y x

as x = y2.

For rotation about

the x-axis, we see that

a typical shell has

radius y, circumference

2πy, and height 1 - y2.

34.
CYLINDRICAL SHELLS METHOD Example 3

So, the volume is: 1

V 2 y 1 y 2 dy

0

1

3

2 ( y y )dy

0

2 4 1

y y

2

2 4 0 2

In this problem, the disk method was simpler.

So, the volume is: 1

V 2 y 1 y 2 dy

0

1

3

2 ( y y )dy

0

2 4 1

y y

2

2 4 0 2

In this problem, the disk method was simpler.

35.
CYLINDRICAL SHELLS METHOD Example 4

Find the volume of the solid obtained by

rotating the region bounded by y = x - x2

and y = 0 about the line x = 2.

Find the volume of the solid obtained by

rotating the region bounded by y = x - x2

and y = 0 about the line x = 2.

36.
CYLINDRICAL SHELLS METHOD Example 4

The figures show the region and a cylindrical

shell formed by rotation about the line x = 2,

which has radius 2 - x, circumference

2π(2 - x), and height x - x2.

The figures show the region and a cylindrical

shell formed by rotation about the line x = 2,

which has radius 2 - x, circumference

2π(2 - x), and height x - x2.

37.
CYLINDRICAL SHELLS METHOD Example 4

So, the volume of the solid is:

0

V 2 2 x x x dx2

1

0

2 x 3 x 2 x dx

3 2

1

4 1

x 3 2

2 x x

4 0 2

So, the volume of the solid is:

0

V 2 2 x x x dx2

1

0

2 x 3 x 2 x dx

3 2

1

4 1

x 3 2

2 x x

4 0 2