1. 6.3 Volumes By Cylindrical Shells This is an alternate method for finding the volume of a solid of revolution that uses cylindrical shells. The strip is parallel to the axis of revolution. This method is sometimes easier to use than the methods discussed in Section 6.2. Volume of Shell height average radius thickness

2. Volume: The Shell Method axis of revolution c d r(y)r(y) h(y)h(y) Plane Region Horizontal Strip Solid of Revolution Typical Shell Volume of Typical Shell where r(y) = average radius h(y)= height = thickness Volume of Solid

3. Examples 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. Example One: Example Two: about the x-axis about x = -2

4. Solutions: Example One 0 1 x y 1 r(y)r(y) h(y)h(y) By Shell Method: x y

5. Solutions: Example Two Points of Intersection: Vertex of parabola 1: (2, 4)Vertex of parabola 2: (2, 8) 024 2 8 x = -2 h(x)h(x)r(x)r(x) By Shell Method: xx y x = -2 y

6. Example: Shell Method Preferable Find the exact volume of the solid formed by revolving the region bounded by the curves: about the y-axis Methods: 1.Disc/Washer – strip is perpendicular to axis of revolution 2.Shell Method- strip is parallel to axis of revolution

7. Solutions: Example by Disc/Washer Method 01 1 2 r x y (1, 2) Note: We found that the disc/washer method requires two integrals to determine the volume of the solid.

8. Solutions: Example by Shell Method 01 1 2 x y (1, 2) Note: We can see that the shell method requires only one integral to find the volume.