Chapter 6 | Applications of Integration
661
revolution formed by revolving Q around the x -axis is given by V = ∫ c d ⎛ ⎝ 2 πyg ( y ) ⎞ ⎠ dy .
Example 6.14 The Method of Cylindrical Shells for a Solid Revolved around the x -axis Define Q as the region bounded on the right by the graph of g ( y ) =2 y and on the left by the y -axis for y ∈ [0, 4]. Find the volume of the solid of revolution formed by revolving Q around the x -axis.
Solution First, we need to graph the region Q and the associated solid of revolution, as shown in the following figure.
Figure 6.31 (a) The region Q to the left of the function g ( y ) over the interval [0, 4]. (b) The solid of revolution generated by revolving Q around the x -axis.
Label the shaded region Q . Then the volume of the solid is given by V = ∫ c d ⎛ ⎝ 2 πyg ( y ) ⎞ ⎠ dy = ∫ 0 4 ⎛ ⎝ 2 πy ⎛ ⎝ 2 y ⎞ ⎠ ⎞ ⎠ dy =4 π ∫ 0 4
y 3/2 dy
⎤ ⎦ ⎥ |
⎡ ⎣ ⎢ 2 y 5/2 5
0 4
= 256 π 5
units 3 .
=4 π
6.14 Define Q as the region bounded on the right by the graph of g ( y ) =3/ y and on the left by the y -axis for y ∈ [1, 3]. Find the volume of the solid of revolution formed by revolving Q around the x -axis.
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