Chapter 6 | Applications of Integration
649
the right by the line x = b . Then, the volume of the solid of revolution formed by revolving R around the x -axis is given by (6.5) V = ∫ a b π ⎡ ⎣ ⎛ ⎝ f ( x ) ⎞ ⎠ 2 − ⎛ ⎝ g ( x ) ⎞ ⎠ 2 ⎤ ⎦ dx .
Example 6.10 Using the Washer Method
Find the volume of a solid of revolution formed by revolving the region bounded above by the graph of f ( x ) = x and below by the graph of g ( x ) =1/ x over the interval [1, 4] around the x -axis.
Solution The graphs of the functions and the solid of revolution are shown in the following figure.
Figure 6.23 (a) The region between the graphs of the functions f ( x ) = x and g ( x ) =1/ x over the interval [1, 4]. (b) Revolving the region about the x -axis generates a solid of revolution with a cavity in the middle.
We have
b
⎡ ⎣ ⎛
⎤ ⎦ dx
V = ∫
⎞ ⎠ 2 −
⎞ ⎠ 2
⎛ ⎝ g ( x )
⎝ f ( x )
π
a
4 ⎡
2 ⎤
⎤ ⎦ | 1 4
⎡ ⎣ x 3
⎛ ⎝ 1 x
⎞ ⎠
= π ∫
= 81 π
1 x
⎣ x 2 −
⎦ dx = π
3 .
3 +
4 units
1
6.10 Find the volume of a solid of revolution formed by revolving the region bounded by the graphs of f ( x ) = x and g ( x ) =1/ x over the interval [1, 3] around the x -axis.
As with the disk method, we can also apply the washer method to solids of revolution that result from revolving a region around the y -axis. In this case, the following rule applies.
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