The student should be familiar with the concept of using the definite integral as a summation, and should also be familiar with the disk and washer methods for finding the volume of a solid of revolution.
This video develops the concept and integral formula for calculating the volume of a solid of revolution using cylindrical shells.
Source: self-created video
This video walks through several examples of applying the shell method and contains instruction on how to set up the shell method for various situations.
Source: self-created video
1. Use both disk (or washer) AND shell method to find the volume of the solid of revolution if the region R is revolved about the given axis.
R is bounded by 
, the x-axis and x = 4.
a. Revolve R about the y-axis. (201.062)
b. Revolve R about the line x = 7. (268.083)
c. Revolve R about the line y = 10. (509.357)
d. Revolve R about the x-axis. (160.85)
2. The first quadrant region bounded by 
and y=4 is revolved about the y=axis.
a. Find the volume of the resulting solid. (25.133)
b. A hole is drilled in the solid. The hole is centered along the y-axis. What must the radius of the hole be to remove one-quarter of the volume of the solid? (0.732)
3. R is bounded by 
. Find the volume of the solid generated when R is revolved about the line x = -2. (1.517)
4. A torus (donut) has a cross section with radius 1. The center of the cross section is 4 units from the center of the torus. Use the concept of solid of revolution and the shell method to calculate the volume of this torus. (78.957)
Source: own problems
Paul’s Online Notes
http://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithCylinder.aspx
Master Math Mentor
http://www.mastermathmentor.com/calc/abcalc.ashx
Shell Method Demo Gallery
http://www.mathdemos.org/mathdemos/shellmethod/gallery/gallery.html
Wolfram Alpha
"I really enjoyed the examples of varied axes of rotation. You provide an easy-to-follow explanation when constructing the formula around an axis of rotation that is not the x- or y-axis."
"Great examples in the second video. The first video takes a difficult concept and makes it very manageable and approachable for a student that may be struggling with the concept. "
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"I really enjoyed the progression you used to justify the use of the shell method and the animations, which helped to bring the method to life."