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Chapter 6 | Applications of Integration
Example 6.13 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.
Solution First graph the region R and the associated solid of revolution, as shown in the following figure.
Figure 6.30 (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 πxf ( x ) ⎞
⎠ dx
2 ⎛
2 ⎛
⎝ 2 x 2 − x 3 ⎞
⎛ ⎝ 2 x − x 2 ⎞ ⎠
⎞ ⎠ dx =2 π ∫
= ∫
⎝ 2 πx
⎠ dx
0
0
⎤ ⎦ | 0 2
⎡ ⎣ 2 x 3
x 4 4
= 8 π
3 .
=2 π
3 −
3 units
6.13 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.
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. Rule: The Method of Cylindrical Shells for Solids of Revolution around the x -axis Let g ( y ) be continuous and nonnegative. Define Q as the region bounded on the right by the graph of g ( y ), on the left by the y -axis, below by the line y = c , and above by the line y = d . Then, the volume of the solid of
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