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Volumes of Revolution The Shell Method

Volumes of Revolution The Shell Method. Lesson 7.3. Shell Method. Based on finding volume of cylindrical shells Add these volumes to get the total volume Dimensions of the shell Radius of the shell Thickness of the shell Height. The Shell.

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Volumes of Revolution The Shell Method

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  1. Volumes of RevolutionThe Shell Method Lesson 7.3

  2. Shell Method • Based on finding volume of cylindrical shells • Add these volumes to get the total volume • Dimensions of the shell • Radius of the shell • Thickness of the shell • Height

  3. The Shell • Consider the shell as one of many of a solid of revolution • The volume of the solid made of the sum of the shells dx f(x) f(x) – g(x) x g(x)

  4. Try It Out! • Consider the region bounded by x = 0, y = 0, and

  5. Hints for Shell Method • Sketch the graph over the limits of integration • Draw a typical shell parallel to the axis of revolution • Determine radius, height, thickness of shell • Volume of typical shell • Use integration formula

  6. Rotation About x-Axis • Rotate the region bounded by y = 4x and y = x2 about the x-axis • What are the dimensions needed? • radius • height • thickness thickness = dy radius = y

  7. Rotation About Noncoordinate Axis • Possible to rotate a region around any line • Rely on the basic concept behind the shell method g(x) f(x) x = a

  8. Rotation About Noncoordinate Axis • What is the radius? • What is the height? • What are the limits? • The integral: r g(x) f(x) a – x x = c x = a f(x) – g(x) c < x < a

  9. r = 2 - x 4 – x2 Try It Out • Rotate the region bounded by 4 – x2 , x = 0 and, y = 0 about the line x = 2 • Determine radius, height, limits

  10. Try It Out • Integral for the volume is

  11. Assignment • Lesson 7.3 • Page 277 • Exercises 1 – 21 odd

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