Two Slice Directions, Two Useful Descriptions
A solid of revolution can often be described using slices in either of two directions. Slices perpendicular to the axis of rotation produce disks or washers. Slices parallel to the axis produce cylindrical shells. Both descriptions can give the same volume, but their bounds and algebra may differ substantially.
The practical question is not simply which formula you remember. It is which slice direction gives the clearest description of the region. A convenient setup usually has simple bounds, a manageable slice length or radii, and as few separate integrals as possible.
In the formulas below, the radius is a distance from the axis, so it is nonnegative. A disk has no hole. A washer has an outer radius \(R\) and an inner radius \(r\), where \(0\le r\le R\). A shell has radius \(r\), thickness \(d x\) or \(d y\), and height equal to the length of the slice parallel to the axis.
Translate Slice Direction into a Formula
For slices perpendicular to the axis, the cross-sectional area is \(\pi R^2\) for a disk or \(\pi(R^2-r^2)\) for a washer. Integrate that area with respect to the coordinate that changes across those slices.
For slices parallel to the axis, each thin slice rotates into a cylindrical shell. Its volume is approximately circumference times height times thickness. Thus a shell has volume contribution \(2\pi rh\) times its thickness, where \(h\) is the slice length parallel to the axis.
For example, about the \(y\)-axis a vertical slice at \(x\) has shell radius \(x\) and thickness \(d x\). If the region runs vertically from \(y=g(x)\) to \(y=f(x)\), its shell height is \(f(x)-g(x)\), provided \(f(x)\ge g(x)\). A horizontal slice about that same axis instead gives a disk or washer, with radii measured horizontally from the \(y\)-axis.
Why the Shell Formula Works
Consider a thin vertical or horizontal strip parallel to the axis. When it rotates, it sweeps out a thin cylindrical shell. Suppose its radius is \(r\), its height is \(h\), and its radial thickness is \(\Delta r\). The exact volume of the resulting cylindrical layer is the difference between two cylinders:
The first term is circumference \(2\pi r\), multiplied by height \(h\) and thickness \(\Delta r\). When \(r>0\), the ratio of the second term to the first is \(\Delta r/(2r)\), so the second term is small relative to the first as \(\Delta r\) tends to zero.
That relative comparison does not apply at \(r=0\): there the first term is zero and the extra term is \(\pi h(\Delta r)^2\). But the extra term still tends to zero in absolute size. More precisely, if the shell heights are bounded by \(H\) and a bounded radial interval is divided into strips of widths \(\Delta r_i\), the sum of these extra terms is at most
The total radial width \(\sum_i\Delta r_i\) is fixed, while the largest strip width tends to zero as the partition is refined. Therefore the total extra contribution tends to zero, including when a strip begins at the axis. The leading terms form Riemann sums for \(\int 2\pi rh\,dr\), which justifies the shell integral for continuous slice heights. For piecewise continuous heights, apply the argument on each piece and add the results.
A Practical Choice of Orientation
Before integrating, sketch the region and the axis of rotation. Then consider both directions, if both are workable:
Perpendicular slices give disks or washers; parallel slices give shells.
Check whether each slice runs across the entire region or whether its endpoints change at some value.
Count inverse functions, separate intervals, and complicated radius or height expressions in each setup.
Use that setup, and consider the other one as an independent check if it is reasonably simple.
A longer integral is not automatically worse, and an inverse function is not automatically a problem. The goal is a correct description of every slice, with bounds that match the variable of integration.
Worked Example: Compare Washers and Shells for a Triangular Region
Let \(R\) be the region between \(y=x\) and \(y=2x\), for \(0\le x\le1\), and rotate it about the \(x\)-axis. Compare vertical washers with horizontal shells.
Vertical slices: At a fixed \(x\), the lower curve is \(y=x\) and the upper curve is \(y=2x\). Rotation about the \(x\)-axis gives a washer with outer radius \(2x\) and inner radius \(x\). The bounds are \(0\le x\le1\), so
Horizontal slices: At a fixed \(y\), the inequalities \(x\le y\le2x\) give \(y/2\le x\le y\). The region also has \(x\le1\). For \(0\le y\le1\), the slice therefore runs from \(x=y/2\) to \(x=y\). For \(1\le y\le2\), it runs from \(x=y/2\) to \(x=1\). Rotating these parallel slices about the \(x\)-axis gives shells of radius \(y\), with the indicated horizontal lengths:
Both setups give \(V=\pi\). The washer setup has one integral, while the shell setup requires a split because the right endpoint of a horizontal slice changes at \(y=1\). Here the washers give the shorter setup.
Check: The independently computed shell volume agrees with the washer volume: \(\pi=\pi\).
Worked Example: Rotate a Region Under a Parabola About the Y-Axis
Let \(R\) be bounded by \(y=0\), \(y=x^2\), \(x=0\), and \(x=2\). Rotate it about the \(y\)-axis. Compare vertical shells with horizontal disks.
Vertical slices: A vertical slice has radius \(x\) and height \(x^2-0=x^2\). It generates a shell, so
Horizontal slices: The \(y\)-values range from \(0\) to \(4\), since \(2^2=4\). Solving \(y=x^2\) for nonnegative \(x\) gives \(x=\sqrt y\). Each horizontal slice extends from the \(y\)-axis to the curve, so it generates a disk of radius \(\sqrt y\):
The shell setup uses a polynomial in \(x\), while the disk setup requires expressing \(x\) in terms of \(y\). In this example the inverse is simple, so both choices are convenient.
Check: The two valid orientations agree: \(8\pi=8\pi\).
Worked Example: Avoid a More Complicated Washer Integral
Let \(R\) be bounded by \(y=0\), \(y=\sqrt{x}\), \(x=0\), and \(x=4\). Rotate it about the \(y\)-axis. Compare vertical shells with horizontal washers.
Vertical slices: For \(0\le x\le4\), a slice has radius \(x\) and height \(\sqrt{x}\). Thus
Horizontal slices: The \(y\)-values run from \(0\) to \(2\). From \(y=\sqrt{x}\), we have \(x=y^2\). At a fixed \(y\), the region extends from \(x=y^2\) to \(x=4\). Rotation about the \(y\)-axis gives a washer with outer radius \(4\) and inner radius \(y^2\):
Both methods work, but the shell setup avoids the fourth power in the washer area. The bounds and endpoints of the horizontal slice are essential: the inner radius is \(y^2\), not \(4-y^2\).
Check: The washer and shell calculations agree, each giving \(128\pi/5\).
A Common Mistake: Treating Every Slice as One Piece
In the triangular-region example, it would be incorrect to use horizontal shell height \(y-y/2\) all the way from \(y=0\) to \(y=2\). That expression assumes the right endpoint is \(x=y\). But once \(y>1\), the region ends at \(x=1\), not \(x=y\). The incorrect setup would be
which does not equal the washer result \(\pi\). The correction is to split at \(y=1\), where the slice endpoint changes, and use height \(1-y/2\) for \(1\le y\le2\). Always determine the complete slice endpoints from all region boundaries before writing the integral.
Choosing Efficiently
The orientation that minimizes algebra depends on the particular region. Perpendicular slices may make radii immediately available but require an inverse boundary function. Parallel slices may use the original functions directly but create shell heights that change across the region. Either method is valid when it accounts for the entire solid exactly once.
- Use washers when the outer and inner radii are easy to express in the perpendicular coordinate.
- Use shells when the radius and parallel slice length are easy to express in the perpendicular coordinate.
- Look for slice endpoints that change, and split the integral wherever the description changes.
- When both setups are simple, agreement between them is a useful check on the bounds and radii.
Check Your Understanding
For each question, identify the orientation and explain which quantities determine the integral.
- For rotation about the \(y\)-axis, which slice direction produces washers, and which produces shells?
- For rotation about the \(x\)-axis, what are the radius and height of a shell formed by a vertical slice at \(x\) through the region between \(y=g(x)\) and \(y=f(x)\), where \(f(x)\ge g(x)\ge0\)?
- Why must the shell integral be split if the endpoints of parallel slices change at an interior value?
- In the region under \(y=\sqrt{x}\) for \(0\le x\le4\), what are the inner and outer radii of a horizontal washer about the \(y\)-axis?
- Why does the shell derivation remain valid for a strip at radius zero even though the relative-error comparison does not apply there?
