Choosing Slice Orientation for a Shell Integral

Why Slice Orientation Matters

A cylindrical-shell integral begins with thin strips of a planar region. Rotating a strip about an axis parallel to it sweeps out a thin cylindrical shell. The strip’s orientation determines whether its distance from the axis and its length can be described simply.

The axis determines the basic choice: vertical strips run parallel to a vertical axis, and horizontal strips run parallel to a horizontal axis. The region then determines how convenient those strips are to describe. A good setup has boundaries that are single-valued in the slice direction, a simple expression for the strip length, and as few interval splits as the geometry allows.

Match the Strips to the Axis

For rotation about the vertical line \(x=c\), a vertical strip at \(x\) runs parallel to the axis. Its shell radius is the horizontal distance \(|x-c|\), its height is the strip’s vertical length, and its thickness is \(dx\). If the region is between a top curve \(y=f(x)\) and a bottom curve \(y=g(x)\), the height is \(f(x)-g(x)\), provided \(f(x)\ge g(x)\) throughout the interval.

For rotation about the horizontal line \(y=d\), use horizontal strips. If a strip at \(y\) has length \(h(y)\) from its left boundary to its right boundary, its radius is \(|y-d|\), and its thickness is \(dy\). Thus, on an interval where the strip length is given by one expression, the shell integral is

In either orientation, the shell height is a length: top minus bottom for a vertical strip, or right minus left for a horizontal strip. If the region has several separate pieces in a strip, their lengths must be accounted for separately rather than treating empty space as part of the height.

Check How the Region Is Described

Once the strip direction is fixed by the axis, inspect the boundary functions. Vertical strips work naturally when the region can be described using \(x\)-intervals and, for each \(x\), a single bottom and top boundary. Horizontal strips work naturally when the region can be described using \(y\)-intervals and, for each \(y\), a single left and right boundary. These descriptions are often called vertically simple and horizontally simple, respectively.

“Single-valued” refers to the boundary as a function of the strip variable. For example, \(y=f(x)\) gives a single \(y\)-value for each allowed \(x\). A curve such as \(x=y^2\) gives \(x\) as a function of \(y\), which is directly useful for horizontal strips. If the region’s boundary has multiple branches, identify which branch forms each edge of the strip and restrict the interval accordingly.

The top and bottom boundaries, or the left and right boundaries, must also stay in the same order over each interval of integration. If they exchange roles at an intersection, split the integral there and write the height separately on each side. A formula that uses “top minus bottom” is not valid on an interval where those labels have switched.

A Derivation of the Shell Setup

Consider a region lying to one side of the vertical axis \(x=c\), with one vertical strip at each \(x\in[a,b]\). Suppose the strip has continuous height \(h(x)\), and the region does not cross the axis. For a small width \(dx\), rotating the strip through a full turn produces a thin shell with radius \(r(x)=|x-c|\), height \(h(x)\), and thickness \(dx\).

A cylinder of radius \(r\) and height \(h\) has volume \(\pi r^2h\). A shell of thickness \(\Delta x\) can be viewed as the difference between two nearby cylinders. If its inner and outer radii are \(r\) and \(r+\Delta x\), respectively, its volume is

After dividing by \(\Delta x\), the first-order contribution per unit thickness is \(2\pi rh\); the term \(\pi(\Delta x)^2h\) becomes negligible as the thickness tends to zero. Summing these shell contributions over a partition of \([a,b]\) and taking the limit gives the integral.

This reasoning requires that the shells represent the solid without counting any part more than once. It also assumes the radii and heights are integrable over the interval; continuous, piecewise-defined boundaries satisfy this condition. The horizontal-strip formula follows by the same argument, using \(dy\) for thickness and distance \(|y-d|\) for radius.

Worked Example: Vertical Shells About the \(y\)-Axis

Worked Example: Horizontal Shells About the \(x\)-Axis

Worked Example: Split Where the Boundaries Switch

A Common Mistake: Counting Shells Twice

Suppose the rectangle \(-1\le x\le1,\ 0\le y\le1\) is rotated about the \(y\)-axis. A tempting but incorrect setup is to use vertical strips across the whole rectangle and write

This doubles the correct volume. A strip at \(x\) and a strip at \(-x\) sweep out the same cylindrical shell, so the integral counts every shell twice. The solid is a cylinder of radius \(1\) and height \(1\), whose volume is \(\pi\). A non-overlapping shell setup uses only one side of the rectangle:

The correction is not to ignore the distance from the axis. It is to ensure that each shell is included exactly once. When the region crosses the axis, check explicitly whether strips on opposite sides generate the same shells before setting up an integral.

A Practical Decision Process

This process helps distinguish a genuinely simple shell setup from one that only appears simple when the region’s changing boundaries or possible overlap are overlooked. If the required strip direction produces several branches or many pieces, keep the geometry explicit and divide the integral wherever its description changes.