Volumes by Cylindrical Shells (Silindirik Kabuk ile Hacim Bulma)

Transcription

1 APPLICATIONS OF INTEGRATION Volumes by Cylindrical Shells (Silindirik Kabuk ile Hacim Bulma) In this section, we will learn: How to apply the method of cylindrical shells to find out the volume of a solid.

2 VOLUMES BY CYLINDRICAL SHELLS Bazı hacim problemlerini önceki yöntemlerle (Disk, Pul) ele almak zordur.

3 VOLUMES BY CYLINDRICAL SHELLS Let s consider the problem of finding the volume of the solid obtained by rotating about the y-axis the region bounded by y = x - x 3 and y =.

4 VOLUMES BY CYLINDRICAL SHELLS 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 = x - x 3 for x in terms of y. That s not easy.

5 VOLUMES BY CYLINDRICAL SHELLS Fortunately, there is a method the method of cylindrical shells that is easier to use in such a case.

6 5 4 3 y = x + VOLUMES BY CYLINDRICAL SHELLS ( ) 5 Find the volume 4 ( y of ) the dy + region 4 bounded by y = x +, x =, 5 and y = ( revolved 5 y) dy + about 4 the y- axis. 5 5y y + 4 We can use the washer method if we split 5 it into two parts: y = x x= y 5 ( ( ) ) y dy + outer radius inner radius Japanese Spider Crab Georgia Aquarium, Atlanta thickness of slice cylinder =

7 5 4 3 y = x + Here is another way we could approach this problem: cross section Dikey bir dilim alır ve y ekseni etrafında döndürürsek bir silindir elde ederiz. Tüm silindirleri birlikte eklersek, orijinal nesneyi yeniden yapılandırabiliriz.

8 5 4 3 y = x + cross section The volume of a thin, hollow cylinder is given by: Lateral surface area of cylinder thickness = circumference height thickness = rhthickness r is the x value of the function. h is the y value of the function. ( ) = x x + r dx h thickness circumference thickness is dx.

9 5 4 3 y = x + This is called the shell method because we use cylindrical shells. cross section = rhthickness ( ) = x x + r dx h thickness circumference If we add all the cylinders from the smallest to the largest: ( ) x x + 3 x + x + 4 x dx x 4 dx 4 +

10 a b x i

11 CYLINDRICAL SHELLS METHOD Here s the best way to remember the formula. Think of a typical shell, cut and flattened, with radius x, circumference πx, height f(x), and thickness x or dx: b a ( ) ( ) x f x dx circumference height thickness

12 CYLINDRICAL SHELLS METHOD Example Let s consider the problem of finding the volume of the solid obtained by rotating about the y-axis the region bounded by y = x - x 3 and y =.

13 VOLUMES BY CYLINDRICAL SHELLS 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 = x - x 3 for x in terms of y. That s not easy.

14 CYLINDRICAL SHELLS METHOD Example So, by the shell method, the volume is: V = x x x dx ( )( 3 ) 3 4 = x ( x x ) dx ( ) = x ( ) = 8 = x

15 CYLINDRICAL SHELLS METHOD Example The figure shows a computer-generated picture of the solid whose volume we computed in the example.

16 When the strip is parallel to the axis of rotation, use the shell method. Şerit dönme eksenine paralel olduğunda, kabuk yöntemini kullanın. When the strip is perpendicular to the axis of rotation, use the washer method. Şerit dönme eksenine dik olduğunda, pul yöntemini kullanın.

17 Example Find the volume generated when this shape is revolved about the y axis y = x x + 9 ( 6) We can t solve for x, so we can t use a horizontal slice directly.

18 If we take a vertical slice and revolve it about the y-axis we get a cylinder. Shell method: y = x x + 9 ( 6) Lateral surface area of cylinder thickness =circumference height thickness = r h dx Volume of thin cylinder = r h dx

19 4 3 Volume of thin cylinder 8 = r h dx 4 ( x x x 6 ) + dx 9 r circumference h thickness y = x x + 9 ( 6) = cm

20 The tables below may be helpful: Method Axis of rotation Integrate in Disks and Washers The x-axis The y-axis x y (use dx ) (use ) dy Cylindrical Shells The x-axis The y-axis x y (use dy ) (use ) dx

21 and Method Axis of rotation formula Disks and Washers The x-axis The y-axis Cylindrical Shells The x-axis The y-axis

22 CYLINDRICAL SHELLS METHOD Example 3 Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x.

23 CYLINDRICAL SHELLS METHOD Example 3 The region and a typical shell are shown here. We see that the shell has radius x, circumference πx, and height x - x.

24 CYLINDRICAL SHELLS METHOD Example 3 Thus, the volume of the solid is: V = x x x dx ( )( ) ( 3) = x x dx 3 4 x x = = 3 4 6

25 CYLINDRICAL SHELLS METHOD As the following example shows, the shell method works just as well if we rotate about the x-axis. We simply have to draw a diagram to identify the radius and height of a shell.

26 CYLINDRICAL SHELLS METHOD Example 4

27 CYLINDRICAL SHELLS METHOD Example 3

28 CYLINDRICAL SHELLS METHOD Example 4 So, the volume is: V = y y dy ( )( ) 3 ( ) = y y dy 4 y y = = 4 In this problem, the disk method was simpler.

29 CYLINDRICAL SHELLS METHOD Example 5 Find the volume of the solid obtained by rotating the region bounded by y = x - x and y = about the line x =.

30 CYLINDRICAL SHELLS METHOD Example 5 The figures show the region and a cylindrical shell formed by rotation about the line x =, which has radius - x, circumference π( - x), and height x - x.

31 CYLINDRICAL SHELLS METHOD Example 5 So, the volume of the solid is: V = x x x dx ( )( ) ( 3 ) = x 3x + x dx 4 x 3 = x + x = 4