Chap6_Sec3 - 6 APPLICATIONS OF INTEGRATION APPLICATIONS...

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APPLICATIONS OF INTEGRATION APPLICATIONS OF INTEGRATION 6
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6.3 Volumes by Cylindrical Shells APPLICATIONS OF INTEGRATION In this section, we will learn: How to apply the method of cylindrical shells to find out the volume of a solid.
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Some volume problems are very difficult to handle by the methods discussed in Section 6.2 VOLUMES BY CYLINDRICAL SHELLS
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Let’s consider the problem of finding the volume of the solid obtained by rotating about the y -axis the region bounded by y = 2 x 2 - x 3 and y = 0. VOLUMES BY CYLINDRICAL SHELLS
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If we slice perpendicular to the y -axis, we get a washer. However, to compute the inner radius and the outer radius of the washer, we would have to solve the cubic equation y = 2 x 2 - x 3 for x in terms of y . That’s not easy. VOLUMES BY CYLINDRICAL SHELLS
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Fortunately, there is a method—the method of cylindrical shells—that is easier to use in such a case. VOLUMES BY CYLINDRICAL SHELLS
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The figure shows a cylindrical shell with inner radius r 1 , outer radius r 2 , and height h . CYLINDRICAL SHELLS METHOD
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Its volume V is calculated by subtracting the volume V 1 of the inner cylinder from the volume of the outer cylinder V 2 . CYLINDRICAL SHELLS METHOD
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Thus, we have: 2 1 2 2 2 1 2 2 2 1 2 1 2 1 2 1 2 1 ( ) ( )( ) 2 ( ) 2 V V V r h r h r r h r r r r h r r h r r π π π π π = - = - = - = + - + = - CYLINDRICAL SHELLS METHOD
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Let ∆ r = r 2 r 1 (thickness of the shell) and (average radius of the shell). Then, this formula for the volume of a cylindrical shell becomes: 2 V rh r π = Formula 1 ( 29 1 2 1 2 r r r = + CYLINDRICAL SHELLS METHOD
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The equation can be remembered as: V = [circumference] [height] [thickness] CYLINDRICAL SHELLS METHOD 2 V rh r π =
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