Volumes By Cylindrical Shells

Objective: To develop another method to find volume without

known cross-sections.

Cylindrical Shells

• 7.3.1 Problem• Let f be continuous and nonnegative on [a, b]

(0 < a < b), and let R be the region that is bounded above by y = f(x), below by the x-axis, and on the sides by the lines x = a and x = b. Find the volume V of the solid of revolution S that is generated by revolving the region R about the y-axis.

Cylindrical Shells

• Sometimes problems of the above type can be solved by the method of disks or washers perpendicular to the y-axis, but when that method is not applicable or the resulting integral is difficult, the method of cylindrical shells, which we will discuss now is a good option.

Cylindrical Shells

• A cylindrical shell is a solid enclosed by two concentric right circular cylinders. The volume V of a cylindrical shell with inner radius r1, outer radius r2, and height h can be written as

V = [area of a cross section][height]

hrr )( 21

22

hrrrr ))(( 1212

)()](2

1[2 1221 rrhrr

Cylindrical Shells

• A cylindrical shell is a solid enclosed by two concentric right circular cylinders. The volume V of a cylindrical shell with inner radius r1, outer radius r2, and height h can be written as

is the average radius and is the thickness

)()](2

1[2 1221 rrhrr

)(2

121 rr

12 rr

][][].[2 thicknessheightradiusaveV

Cylindrical Shells

• When using cylindrical shells, the underlying idea is to divide the interval [a, b] into n subintervals, thereby subdividing the region R into n strips, R1, R2, …,Rn. When the region is revolved around the y-axis, these strips generate “tube-like” solids S1, S2, …, Sn that are nested one inside the other and together comprise the entire solid S. The volume V of the solid can be obtained by adding together the volumes of the tubes; that is,

)(...)()( 21 nSVSVSVV

Cylindrical Shells

• As a rule, the tubes will have curved upper surfaces, so there will be no simple formulas for their volumes. However, if the strips are thin, then we can approximate each strip by a rectangle. These rectangles, when revolved about the y-axis, will produce cylindrical shells whose volumes closely approximate the volumes of the tubes generated by the original strips. We will show that by adding the volumes of the cylindrical shells we can obtain a Riemann Sum that approximates the volume V, and by taking the limit of the Riemann Sums we can obtain an integral for the exact value.

Cylindrical Shells

• To implement this idea, suppose that the kth strip extends from xk-1 to xk and that the width of the strip is . If we let be the midpoint of the interval , and if we construct a rectangle of height over the interval, then revolving this rectangle about the y-axis produces a cylindrical shell of average radius , height , and thickness . From , the volume is

1 kkk xxx*kx ],[ 1 kk xx

)( *kxf

*kx )( *

kxf kx][][].[2 thicknessheightradiusaveV

kkkk xxfxV )(2 **

Cylindrical Shells

• Adding the volumes of the n cylindrical shells yields the following Riemann Sum that approximates the volume V:

kk

n

kk xxfxV

)(2 *

1

*

Cylindrical Shells

• Adding the volumes of the n cylindrical shells yields the following Riemann Sum that approximates the volume V:

• Taking the limit as n increases and the widths of the subintervals approach zero yields the definite integral

kk

n

kk xxfxV

)(2 *

1

*

b

a

k

n

kkk

xdxxxfxxfxV )(2)(2lim

1

**

0max

Volume by Cylindrical Shells

• 7.3.2 Let f be continuous and nonnegative on [a, b] (0 < a < b), and let R be the region that is bounded above by y = f(x), below by the x-axis, and on the sides by the lines x = a and x = b. Then the volume V of the solid of revolution that is generated by revolving the region R about the y-axis is given by

b

a

dxxxfV )(2

Example 1a

• Find the volume of the solid generated when the region enclosed between the curves and the x-axis is revolved about the y-axis.

,4,1, xxxy

Example 1a

• Find the volume of the solid generated when the region enclosed between the curves and the x-axis is revolved about the y-axis.

• The “right curve” is always x = 4. However, the “left curve” is x = 1 from 0-1, and it is x = y2 from 1-2.

,4,1, xxxy

Example 1a

• Find the volume of the solid generated when the region enclosed between the curves and the x-axis is revolved about the y-axis.

• The “right curve” is always x = 4. However, the “left curve” is x = 1 from 0-1, and it is x = y2 from 1-2.

,4,1, xxxy

15)14(1

0

22 dy

8.9))(4(2

1

222 dyy

5

124

Example 1b

• Use cylindrical shells to find the volume of the solid generated when the region enclosed between the curves and the x-axis is revolved about the y-axis.

,4,1, xxxy

Example 1b

• Use cylindrical shells to find the volume of the solid generated when the region enclosed between the curves and the x-axis is revolved about the y-axis.

,4,1, xxxy

389.975

1242

4

1

dxxxV

Example 2a

• Find the volume of the solid generated when the region R in the first quadrant enclosed between y = x and y = x2 is revolved about the y-axis.

Example 2a

• Find the volume of the solid generated when the region R in the first quadrant enclosed between y = x and y = x2 is revolved about the y-axis.

1

0

22 523.])[( dyyy

Example 2b

• Use cylindrical shells to find the volume of the solid generated when the region R in the first quadrant enclosed between y = x and y = x2 is revolved about the y-axis.

Example 2b

• Use cylindrical shells to find the volume of the solid generated when the region R in the first quadrant enclosed between y = x and y = x2 is revolved about the y-axis.

• As you can see, the cross section of R parallel to the y-axis generates a cylindrical surface of height x – x2

and radius x.

1

0

2 523.)(2 dxxxxV

Example 3

• Use cylindrical shells to find the volume of the solid generated when the region R under y = x2 over the interval [0, 2] is revolved about the line y = -1.

Example 3

• Use cylindrical shells to find the volume of the solid generated when the region R under y = x2 over the interval [0, 2] is revolved about the line y = -1.

• As illustrated in the figure, at each y in the interval 0 < y < 4, the cross section of R

parallel to the x-axis generates a cylindrical surface of height and radius y + 1.

y2

Example 3

• Use cylindrical shells to find the volume of the solid generated when the region R under y = x2 over the interval [0, 2] is revolved about the line y = -1.

• As illustrated in the figure, at each y in the interval 0 < y < 4, the cross section of R

parallel to the x-axis generates a cylindrical surface of height and radius y + 1.

y2

861.36)2)(1(24

0

dyyyV

Homework

• Page 464• 5, 7, 13, 15, 23