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V shell 2 π f ( x i * ) x i * Δ x .

Another way to think of this is to think of making a vertical cut in the shell and then opening it up to form a flat plate ( [link] ).

This figure has two images. The first is labeled “a” and is of a hollow cylinder around the y-axis. On the front of this cylinder is a vertical line labeled “cut line”. The height of the cylinder is “y=f(x)”. The second figure is labeled “b” and is a shaded rectangular block. The height of the rectangle is “f(x*), the width of the rectangle is “2pix*”, and the thickness of the rectangle is “delta x”.
(a) Make a vertical cut in a representative shell. (b) Open the shell up to form a flat plate.

In reality, the outer radius of the shell is greater than the inner radius, and hence the back edge of the plate would be slightly longer than the front edge of the plate. However, we can approximate the flattened shell by a flat plate of height f ( x i * ) , width 2 π x i * , and thickness Δ x ( [link] ). The volume of the shell, then, is approximately the volume of the flat plate. Multiplying the height, width, and depth of the plate, we get

V shell f ( x i * ) ( 2 π x i * ) Δ x ,

which is the same formula we had before.

To calculate the volume of the entire solid, we then add the volumes of all the shells and obtain

V i = 1 n ( 2 π x i * f ( x i * ) Δ x ) .

Here we have another Riemann sum, this time for the function 2 π x f ( x ) . Taking the limit as n gives us

V = lim n i = 1 n ( 2 π x i * f ( x i * ) Δ x ) = a b ( 2 π x f ( x ) ) d x .

This leads to the following rule for the method of cylindrical shells.

Rule: the method of cylindrical shells

Let f ( x ) be continuous and nonnegative. Define R as the region bounded above by the graph of f ( x ) , below by the x -axis , on the left by the line x = a , and on the right by the line x = b . Then the volume of the solid of revolution formed by revolving R around the y -axis is given by

V = a b ( 2 π x f ( x ) ) d x .

Now let’s consider an example.

The method of cylindrical shells 1

Define R as the region bounded above by the graph of f ( x ) = 1 / x and below by the x -axis over the interval [ 1 , 3 ] . Find the volume of the solid of revolution formed by revolving R around the y -axis .

First we must graph the region R and the associated solid of revolution, as shown in the following figure.

This figure has three images. The first is a solid that has been formed by rotating the curve y=1/x about the y-axis. The solid begins on the x-axis and stops where y=1. The second image is labeled “a” and is the graph of y=1/x in the first quadrant. Under the curve is a shaded region labeled “R”. The region is bounded by the curve, the x-axis, to the left at x=1 and to the right at x=3. The third image is labeled “b” and is half of the solid formed by rotating the shaded region about the y-axis.
(a) The region R under the graph of f ( x ) = 1 / x over the interval [ 1 , 3 ] . (b) The solid of revolution generated by revolving R about the y -axis .

Then the volume of the solid is given by

V = a b ( 2 π x f ( x ) ) d x = 1 3 ( 2 π x ( 1 x ) ) d x = 1 3 2 π d x = 2 π x | 1 3 = 4 π units 3 .
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Define R as the region bounded above by the graph of f ( x ) = x 2 and below by the x -axis over the interval [ 1 , 2 ] . Find the volume of the solid of revolution formed by revolving R around the y -axis .

15 π 2 units 3

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The method of cylindrical shells 2

Define R as the region bounded above by the graph of f ( x ) = 2 x x 2 and below by the x -axis over the interval [ 0 , 2 ] . Find the volume of the solid of revolution formed by revolving R around the y -axis .

First graph the region R and the associated solid of revolution, as shown in the following figure.

This figure has two graphs. The first graph is labeled “a” and is the curve f(x)=2x-x^2. It is an upside down parabola intersecting the x-axis at the origin ant at x=2. Under the curve the region in the first quadrant is shaded and is labeled “R”. The second figure is a graph of the same curve. On the graph is a solid that is formed by rotation the region from “a” about the y-axis.
(a) The region R under the graph of f ( x ) = 2 x x 2 over the interval [ 0 , 2 ] . (b) The volume of revolution obtained by revolving R about the y -axis .

Then the volume of the solid is given by

V = a b ( 2 π x f ( x ) ) d x = 0 2 ( 2 π x ( 2 x x 2 ) ) d x = 2 π 0 2 ( 2 x 2 x 3 ) d x = 2 π [ 2 x 3 3 x 4 4 ] | 0 2 = 8 π 3 units 3 .
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Define R as the region bounded above by the graph of f ( x ) = 3 x x 2 and below by the x -axis over the interval [ 0 , 2 ] . Find the volume of the solid of revolution formed by revolving R around the y -axis .

8 π units 3

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As with the disk method and the washer method, we can use the method of cylindrical shells with solids of revolution, revolved around the x -axis , when we want to integrate with respect to y . The analogous rule for this type of solid is given here.

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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