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

2. Volumes by Cylindrical Shells Erickson6.3 Volumes by Cylindrical Shells2  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:

3. Methods of Shells Erickson6.3 Volumes by Cylindrical Shells3  In this case we will use the shells method as seen below.

4. Method of Cylindrical Shells 6.3 Volumes by Cylindrical Shells4 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: Erickson

5. Method of Cylindrical Shells Erickson6.3 Volumes by Cylindrical Shells5  Here is another example of how we can create a cylindrical shell.

6. Definition of Volume (Shell) 6.3 Volumes by Cylindrical Shells6 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 Circumference Height Thickness Erickson

7. Definition of Volume (Shell) 6.3 Volumes by Cylindrical Shells7 The volume of a solid is obtained by rotating about the x-axis the region under the curve x = f (y) from c to d is where 0  c < d Circumference Height Thickness Erickson

8. Example 1 6.3 Volumes by Cylindrical Shells8  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? Erickson

9. Example 2 6.3 Volumes by Cylindrical Shells9 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.  a.  b. Erickson

10. Example 3 6.3 Volumes by Cylindrical Shells10 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.  a.  b. Erickson