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Exploring Volumes by Cylindrical Shells: Mathematical Techniques for Rotating Solids

Learn how to calculate volumes of solids obtained by rotating regions using cylindrical shells. Discover the methods of disks and shells, understand the difference, and practice with examples. Unleash your imagination in visualizing rotating shapes. Developed by Lila F. Roberts, this educational demo is a valuable resource for math enthusiasts.

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Exploring Volumes by Cylindrical Shells: Mathematical Techniques for Rotating Solids

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  1. VOLUMES Volume = Area of the base X height

  2. VOLUMES BY CYLINDRICAL SHELLS Cylinder Cylinder Shell

  3. VOLUMES BY CYLINDRICAL SHELLS solid obtained by rotating a region What is the shape of a solid if you rotate the Square about the y-axis What is the shape of a solid if you rotate the triangle about the y-axis

  4. VOLUMES BY CYLINDRICAL SHELLS solid obtained by rotating a region Rotating axis

  5. VOLUMES solid obtained by rotating a region use your imagination

  6. VOLUMES BY CYLINDRICAL SHELLS Example: Find the volume of the solid obtained by rotating the region enclosed by the curves y=sin(x) and y=0 about the y-axis . Find the volume of the resulting solid. Disk Method Shell Method The method of shells is fundamentally different from the method of disks.  The method of disks involves slicing the solid perpendicular to the axis of revolution to obtain the disks.  However, the method of shells fills the solid with cylindrical shells This demo was developed by Lila F. Roberts College of Information & Mathematical Sciences Clayton State University Morrow, GA 30260

  7. VOLUMES BY CYLINDRICAL SHELLS

  8. VOLUMES BY CYLINDRICAL SHELLS The volume is given by Find the surface area

  9. VOLUMES BY CYLINDRICAL SHELLS step1 Graph and Identify the region Draw a line parallel to the rotating line at the point x step2 step3 Rotate this line about the rotating line step4 in terms of Find: step4 Find: step5 The volume is given by Note: rotating line is y-axis  dx and we draw a parallel line to y-axis

  10. VOLUMES BY CYLINDRICAL SHELLS

  11. Final T-131

  12. VOLUMES T-131

  13. VOLUMES BY CYLINDRICAL SHELLS T-122 1 Graph and Identify the region Draw a line parallel to the rotating line at the point y 2 3 Rotate this line about the rotating line 4 in terms of Find: 5 Find: 6 The volume is given by Note: rotating line is x-axis  dy and we draw a parallel line to x-axis

  14. VOLUMES T-102

  15. VOLUMES BY CYLINDRICAL SHELLS

  16. VOLUMES BY CYLINDRICAL SHELLS rotating line Parallel to y-axis CYLINDRICAL SHELLS The volume is given by Find the surface area rotating line Parallel to x-axis CYLINDRICAL SHELLS The volume is given by Find the surface area

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