1. Lecture 1 – Volumes Area – the entire 2-D region was sliced into strips Before width(  x) was introduced, only dealing with length ab f(x) Volume – same concept, 3-D solid is sliced into strips Before width is introduced, only dealing with 2-D area 1

2. Solids of revolution Formed when revolving a region around a given line (axis). Follow one slice of the region (strip) as is gets swept out twice. 1 st sweep: generates an area. 2 nd sweep: generates a volume. Infinite number of disks used, hence Riemann sum turns to integral. Revolve over x-axis 2 f(x) x

3. Example 1 Find the volume of the solid generated by revolving the region bounded the given curves around the x-axis. 3 f(x) x 12

4. Example 2 Find the volume of the solid generated by revolving the region bounded the given curves around the x-axis. 4 f(x) x 123

5. Washer method What happens when the region is revolved about a line but there is a gap between the two? Then the solid generated has a hole. Revolve over x-axis 5 f(x) x

6. Example 3 Find the volume of the previous solid. 6

7. Example 4 Find the volume of the solid generated by revolving the region from Example 1 around the line y = -1. 7 f(x) x 12

8. Lecture 2 – More Volumes 8 Find the volume of the solid generated by revolving the region bounded the given curves around the y-axis. Example 4

9. 9 Find the volume of the solid generated by revolving the region bounded the given curves around the x = 5. Example 6 5

10. What if you prefer to figure out everything in terms of the x-axis? How can volume work if revolution is around vertical axis? Disks(Washers) – when strip is perpendicular to axis of revolution Shells are created when strip is parallel to axis of revolution. 10

11. 11 Find the volume of the solid generated by revolving the region bounded the given curves around the y-axis using shells. Example 7

12. 12 Find the volume of the solid generated by revolving the region bounded the given curves around the x = 5 using shells. Example 8 5