Lecture Notes: Volume by Shell Method

Key Concepts

• Volume by Shell Method: A technique for finding the volume of a solid of revolution where the solid is generated by rotating around an axis.
• Difference from Washer Method: In the shell method, the axis of rotation is parallel to the shells.

Shell Method Description

• Axis of Rotation: For the shell method, the axis of rotation is parallel to the shell.
• Region of Interest: Defined by function y = f(x) above the x-axis and bounded within x = A and x = B.
• Rotation: Revolves around the y-axis.

Rectangle and Shell

• Rectangle Setup: Draw a rectangle, parallel to the axis of rotation, which upon rotation forms a shell.
• Thickness: The thickness of the shell is Δx = (x_i - x_(i-1)).

Calculating Volume

• Volume of One Shell: Volume is calculated as 2π times the average radius times the height times the thickness (Δx).
• Formula for One Shell:
  • V_shell = 2π * x_i_star * f(x_i_star) * Δx
  • Where x_i_star is a midpoint value.

Integral for Volume

• Infinite Number of Shells: To find the total volume, take the limit as n approaches infinity.
• Definite Integral:
  • Volume = ∫ from A to B of 2π * x * f(x) dx

Example Problems

Example 1

• Function: f(x) = 1/√x
• Bounds: Bounded by x = 1 and x = 4
• Rotation: Around the y-axis
• Steps:
  1. Set up integral using formula 2π ∫ from 1 to 4 of x * (1/√x) dx
  2. Simplify and solve the integral
  3. Calculate volume: 28π/3 cubic units

Example 2

• Function: f(y) = √x, bounds 0 to 4
• Rotation: Around x-axis
• Steps:
  1. Rewrite function in terms of y: x = y²
  2. Set up integral 2π ∫ from 0 to 2 of y * (4 - y²) dy
  3. Calculate volume: 8π cubic units

Conclusion

• Key Takeaway: When using the shell method, remember that the shell is parallel to the axis of rotation. The calculation involves integrating the product of radius and height over the interval of rotation.
• Practice: Draw and visualize the rotational shells for better understanding. This method is useful for complex shapes where the washer method may be cumbersome.