A cylindrical shell may be regarded as the outer skin of a cylinder. Its volume is the volume of the
rectangular solid formed when this skin is peeled from the cylinder and flattened out. As an example,
consider the volume of the solid of revolution formed when the region bounded by the two curves
seen in Figure N7–17 is revolved around the y-axis. We think of the rectangular strip of the region at
the left as generating the shell, ΔV (an element of the volume), shown at the right.
FIGURE N7–17
This shell’s radius, r, is the distance from the axis to the rectangular strip, and its height is the
height of the rectangular strip, h. When the shell is unwound and flattened to form a rectangular solid,
the length of the solid is the circumference of the cylinder, 2πr, its height is the height of the cylinder,
h, and its thickness is the thickness of the rectangular strip, Δx. Thus:
‡Examples 10–12 involve finding volumes by the method of shells. Although shells are not included in the Topic Outline, we include this
method here because it is often the most efficient (and elegant) way to find a volume. No question requiring shells will appear on the
AP exam.
EXAMPLE 10
Find the volume of the solid generated when the region bounded by y = x^2 , x = 2, and y = 0 is rotated
about the line x = 2. See Figure N7–18.SOLUTION:
About x = 2.
Shell.