3c3-Volums-CylinShells_Stu - Volumes by Cylindrical Shells...

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Some volume problems are very difficult to handle by the methods of Section 6.2. For instance, let’s consider the problem of finding the volume of the solid obtained by rotating about the -axis the region bounded by and . (See Figure 1.) If we slice perpendicular to the y -axis, we get a washer. But to compute the inner radius and the outer radius of the washer, we would have to solve the cubic equation for x in terms of y ; that’s not easy. Fortunately, there is a method, called the method of cylindrical shells , that is easier to use in such a case. Figure 2 shows a cylindrical shell with inner radius , outer radius , and height . Its volume is calculated by subtracting the volume of the inner cylinder from the volume of the outer cylinder: If we let (the thickness of the shell) and (the average radius of the shell), then this formula for the volume of a cylindrical shell becomes and it can be remembered as Now let be the solid obtained by rotating about the -axis the region bounded by [where ], and , where . (See Figure 3.) We divide the interval into n subintervals of equal width and let be the midpoint of the i th subinterval. If the rectangle with base and height is rotated about the y -axis, then the result is a cylindrical shell with average radius , height , and thickness (see Figure 4), so by Formula 1 its volume is Therefore, an approximation to the volume of is given by the sum of the volumes of these shells: V ± ² n i ± 1 V i ± ² n i ± 1 2 ± x i f ³ x i ´ ² x S V V i ± ³ 2 x i ´µ f ³ x i ´¶ ² x ² x f ³ x i ´ x i f ³ x i ´ µ x i ³ 1 , x i x i ² x µ x i ³ 1 , x i µ a , b FIGURE 3 ab x y 0 y=ƒ x y 0 y=ƒ b ´ a µ 0 x ± b y ± 0, x ± a , f ³ x ´ µ 0 y ± f ³ x ´ y S V ± [circumference][height][thickness] V ± 2 rh ² r 1 r ± 1 2 ³ r 2 r 1 ´ ² r ± r 2 ³ r 1 ± 2 r 2 r 1 2 h ³ r 2 ³ r 1 ´ ± ³ r 2 r 1 ´³ r 2 ³ r 1 ´ h ± r 2 2 h ³ r 2 1 h ± ³ r 2 2 ³ r 2 1 ´ h V ± V 2 ³ V 1 V 2 V 1 V h r 2 r 1 y ± 2 x 2 ³ x 3 y ± 0 y ± 2 x 2 ³ x 3 y FIGURE 1 FIGURE 2 r r™ Îr h y x 0 2 1 y=2≈-˛ x L = ? x R = ? 1 Volumes by Cylindrical Shells
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This approximation appears to become better as . But, from the definition of an inte- gral, we know that Thus, the following appears plausible: The volume of the solid in Figure 3, obtained by rotating about the y -axis the region under the curve from a to b ,is The argument using cylindrical shells makes Formula 2 seem reasonable, but later we will be able to prove it. (See Exercise 47.) The best way to remember Formula 2 is to think of a typical shell, cut and flattened as in Figure 5, with radius x , circumference , height , and thickness or : This type of reasoning will be helpful in other situations, such as when we rotate about lines other than the y -axis.
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3c3-Volums-CylinShells_Stu - Volumes by Cylindrical Shells...

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