Volumes of Solids of Revolution: Method of Cylinders

Introduction

• Focus on calculating volumes of solids of revolution using the method of cylinders (or method of shells).
• Builds on concepts from the previous section which dealt with disks and rings.

Key Concepts

• Method of Cylinders: Uses cylindrical shells to determine volume.
• Formula for area: [ A = 2 \pi (\text{radius})(\text{height}) ]
• Rotation axis impacts the function of area:
  • Vertical axis: area as a function of (x)
  • Horizontal axis: area as a function of (y)
• Limits of integration differ from the method of disks/rings:
  • Use a range that covers one side of the solid.
  • Expanding the radius on one side automatically covers the other side.

Example Applications

Example 1

• Objective: Determine the volume of a solid obtained by rotating the region
  • Bounded by ( y = (x-1)(x-3)^2 ) and the (x)-axis
  • About the (y)-axis
• Method: Demonstrates how the method of cylinders is applied when a solid cannot be approached using rings/disks.

Example 2

• Objective: Calculate the volume of a solid obtained by rotating the region
  • Bounded by ( y = 3x ), ( x = 3 ), ( x = 8 ), and the (x)-axis
  • About the (x)-axis
• Method: Illustrates the calculation involving a horizontal axis of rotation.

Example 3

• Objective: Calculate the volume of a solid obtained by rotating the region
  • Bounded by ( y = 2x - 1 ) and ( y = x - 1 )
  • About the line ( x = 6 )
• Method: Highlights axis of rotation other than the main axes, requiring careful setup.

Example 4

• Objective: Determine the volume of a solid obtained by rotating the region
  • Bounded by ( x = (y-2)^2 ) and ( y = x )
  • About the line ( y = 1 )
• Method: Demonstrates use of non-traditional axes of rotation.

Notes

• Important to choose the correct method based on the problem's axis of rotation.
• Method of cylinders provides flexibility with complex shapes where rings/disks aren't suitable.

Additional Information

• Paul Dawkins' online notes are a primary reference.
• The page was last modified on 11/16/2022.