Count thin cylindrical layers

Try

Roll a thin rectangle of height h into the side of a cylinder of radius r. Its width around the cylinder is 2πr, so a thin layer of radial thickness Δr has approximate volume 2πr h Δr. Identify the three length factors. Unlike a washer, this piece wraps around the axis instead of lying perpendicular to it.

Predict

The region under y=x from x=0 to x=2 rotates about the y-axis. Predict the shell radius and height of the vertical strip at x=1.5. Both are 1.5 in this example, but they describe different distances. Their numerical equality is not a general shell rule.

Experiment

List the factors before constructing the integral.

Object	Expression for this region
Distance to y-axis	radius x
Vertical extent	height x−0=x
Circumference	2πx
Thin volume contribution	2πx² dx

Integration gives 2π∫₀²x²dx=16π/3. Independently slice horizontally: at height y, the region runs from x=y to x=2. Washers give π∫₀²(4−y²)dy=16π/3. Draw both representative slices on the same region and explain why the solid is identical despite different integrands.

Understand

The shell formula accumulates circumference times height times radial thickness. For vertical strips rotating about the y-axis on a nonnegative-x region, it takes the form 2π∫x·height(x)dx. A shifted vertical axis x=c changes the radius to |x−c|. Horizontal strips about a horizontal axis instead use y distances and dy. The orientation of the strip relative to the axis, rather than a memorized variable, determines the method.

For the region under y=1−x on [0,1] rotating about x=−1, radius is x+1 and height is 1−x. Volume is 2π∫₀¹(x+1)(1−x)dx=2π∫₀¹(1−x²)dx=4π/3. A radius of x would describe rotation about a different axis and give a different solid.

The thin-layer expression can be motivated without ignoring finite thickness. An annular layer from radius r to r+Δr with constant height h has volume π[(r+Δr)²−r²]h=2πrhΔr+πh(Δr)². The second term becomes negligible in a suitably refining sum. For changing heights, the integral formalizes the limiting process under appropriate regularity. A finite shell approximation is not an exact identity for a thick curved layer.

Avoid double counting. Rotating the rectangle −1≤x≤1 and 0≤y≤1 about the y-axis creates a cylinder of radius 1 and height 1. Strips at x and −x sweep the same radial layers. Integrating 2π|x| over both halves counts them twice, giving 2π instead of π. Using only the nonnegative half gives the correct volume. Regions crossing an axis require careful analysis of the union of swept layers, not automatic absolute values.

Choose shells when their radius and height are simpler than the inverse expressions needed for washers. Choose washers when cross-sectional radii are simpler. A correct choice is justified by a region sketch and a nonoverlapping partition; method names alone do not determine a valid setup.

Apply

For each practice problem, identify the strip direction, rotation axis, radius, height and differential. Verify one volume by the other slicing method. In the overlap task, explain which geometric layers were counted twice rather than merely dividing the erroneous answer by two without reasoning.

Continue

Next replace a varying function by a constant that produces the same integral. This average-value construction uses interval length as a normalization, with different units from volume.