Chapter 6 | Applications of Integration
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Figure 6.32 (a) The region R between the graph of f ( x ) and the x -axis over the interval [1, 2]. (b) The solid of revolution generated by revolving R around the line x =−1.
Note that the radius of a shell is given by x +1. Then the volume of the solid is given by V = ∫ 1 2 ⎛ ⎝ 2 π ( x +1) f ( x ) ⎞ ⎠ dx = ∫ 1 2 (2 π ( x +1) x ) dx =2 π ∫ 1 2 ⎛ ⎝ x 2 + x ⎞ ⎠ dx =2 π ⎡ ⎣ x 3 3 + x 2 2 ⎤ ⎦ | 1 2 = 23 π 3 units 3 .
6.15 Define R as the region bounded above by the graph of f ( x ) = x 2 and below by the x -axis over the interval [0, 1]. Find the volume of the solid of revolution formed by revolving R around the line x =−2.
For our final example in this section, let’s look at the volume of a solid of revolution for which the region of revolution is bounded by the graphs of two functions. Example 6.16 A Region of Revolution Bounded by the Graphs of Two Functions Define R as the region bounded above by the graph of the function f ( x ) = x and below by the graph of the function g ( x ) =1/ x over the interval [1, 4]. Find the volume of the solid of revolution generated by revolving R around the y -axis.
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