Chap6_Sec3

Chap6_Sec3 - CYLINDRICAL SHELLS METHOD Example 1 (6.3) Find...

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Find the volume of the solid obtained by rotating about the y-axis the region bounded by y = 2x 2 -x 3 and y = 0. We see that a typical shell has radius x , circumference 2 π x , and height f ( x ) = 2 x 2 3 . So, by the shell method, the volume is: Example 1 (6.3) CYLINDRICAL SHELLS METHOD () ( ) 2 23 0 2 34 0 2 45 11 25 0 32 16 55 22 2( 2 ) 2 28 π ππ =− ⎡⎤ ⎣⎦ = Vx x x d x xx x d x
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Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x 2 . The region and a typical shell are shown here. ± We see that the shell has radius x , circumference 2 π x , and height x - x 2 . Thus, the volume of the solid is: Example 2 (6.3) CYLINDRICAL SHELLS METHOD () ( ) 1 2 0 1 23 0 1 34 0 2 2 2 6 Vx x x d x x xd x xx π =− ⎡⎤ = ⎢⎥ ⎣⎦
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Use cylindrical shells to find the volume of the solid obtained by rotating about the x-axis the region under the curve from 0 to 1. ± This problem was solved using disks in Example 2 in Section 6.2 To use shells, we relabel the curve as x = y 2 . ± For rotation about the x -axis, we see that a typical shell has radius
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This note was uploaded on 05/23/2011 for the course MAC 2311 taught by Professor Noohi during the Fall '08 term at FSU.

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Chap6_Sec3 - CYLINDRICAL SHELLS METHOD Example 1 (6.3) Find...

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