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Lecture 1 – Volumes

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

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Lecture 1 – Volumes

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

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

  3. Example 1 Find the volume of the solid generated by revolving the region bounded the given curves around the x-axis. f(x) x 1 2

  4. Example 2 Find the volume of the solid generated by revolving the region bounded the given curves around the x-axis. f(x) x 1 2 3

  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. f(x) Revolve over x-axis x

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

  7. Example 4 f(x) Find the volume of the solid generated by revolving the region from Example 1 around the line y = -1. x 1 2

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

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

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

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

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