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Chapter 6 – Applications of Integration

Chapter 6 – Applications of Integration. 6.3 Volumes by Cylindrical Shells. Volumes by Cylindrical Shells. In the last section we learned how to find volumes of solids by the disk and washer methods. However those methods would not work in the following case:

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Chapter 6 – Applications of Integration

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  1. Chapter 6 – Applications of Integration 6.3 Volumes by Cylindrical Shells 6.3 Volumes by Cylindrical Shells

  2. Volumes by Cylindrical Shells • In the last section we learned how to find volumes of solids by the disk and washer methods. However those methods would not work in the following case: • Consider the solid generated by rotating the shaded area below with respect to y-axis: 6.3 Volumes by Cylindrical Shells

  3. Methods of Shells • In this case we will use the shells method as seen below. 6.3 Volumes by Cylindrical Shells

  4. Method of Cylindrical Shells We let the thickness of the shell be and we also have so then the formula for the volume of a sphere is as follows: 6.3 Volumes by Cylindrical Shells

  5. Method of Cylindrical Shells • Here is another example of how we can create a cylindrical shell. 6.3 Volumes by Cylindrical Shells

  6. Definition of Volume (Shell) The volume of a solid is obtained by rotating about the y-axis the region under the curve y = f (x) from a to b is where 0  a < b CircumferenceHeight Thickness 6.3 Volumes by Cylindrical Shells

  7. Example 1 – Page 444 #2 • Let S be the solid obtained by rotating the region shown in the figure about the y-axis. Sketch a typical cylindrical shell and find its circumference and height. Use shells to find the volume of S. Do you think this method is preferable to slicing? 6.3 Volumes by Cylindrical Shells

  8. Example 2 – Page 437 #18 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. Sketch the region and a typical shell. • 18. • (NIB) 6.3 Volumes by Cylindrical Shells

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