Chapter 6 | Applications of Integration
659
formed by revolving R around the y -axis is given by
(6.6)
b
V = ∫
⎛ ⎝ 2 πxf ( x ) ⎞
⎠ dx .
a
Now let’s consider an example.
Example 6.12 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.
Solution First we must graph the region R and the associated solid of revolution, as shown in the following figure.
Figure 6.29 (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
b
V = ∫
⎛ ⎝ 2 πxf ( x ) ⎞
⎠ dx
a
3 ⎛
⎛ ⎝ 1 x
⎞ ⎠
⎞ ⎠ dx
= ∫ = ∫
⎝ 2 πx
1
3 2 πdx =2 πx | 1
3 =4 π units 3 .
1
6.12 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.
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