Why Check a Volume Two Ways?
A solid of revolution can often be sliced in two different directions. Slices perpendicular to the axis of rotation produce disks or washers; slices parallel to the axis produce cylindrical shells. When both descriptions are practical, their volume integrals should give the same result.
That agreement is useful, but it is more than a second route to an answer. It checks whether the bounds, radii, heights, and slice orientation in each setup describe the same solid. The two integrands generally look different, so matching results do not mean the setups are identical. They mean the two methods have measured the same volume.
The Two Volume Descriptions
For washers, each slice is perpendicular to the axis of rotation. If its outer radius is \(R\), inner radius is \(r\), and thickness is \(dx\) or \(dy\), its cross-sectional area is \(\pi(R^2-r^2)\). The volume is the integral of these areas over the slice interval.
For shells, each slice is parallel to the axis of rotation. A thin shell at radius \(\rho\), with height or length \(h\) and thickness \(d\rho\), has approximate volume \(2\pi\rho h\,d\rho\). Integrating gives the shell volume. The shell radius is the distance from the slice to the axis, not the slice's height or length.
In each method, determine the geometry before integrating. For washers, identify the farthest and nearest distances from the axis; a disk is the special case \(r=0\). For shells, identify the radius and the length of the segment being rotated. Check that these distances are nonnegative and that the slice bounds describe the entire region.
Worked Example: A Right Triangle Rotated About the \(y\)-Axis
Worked Example: A Right Triangle Rotated About the \(y\)-Axis
Let the region be bounded by \(x=0\), \(y=0\), and \(y=2-x\). Rotate it about the \(y\)-axis. Set up and evaluate both methods.
Shells: Use vertical slices, parallel to the \(y\)-axis. The variable runs from \(x=0\) to \(x=2\). A slice at \(x\) is distance \(x\) from the axis, so its shell radius is \(x\). Its height is the top boundary minus the bottom boundary, \((2-x)-0=2-x\). Therefore,
An antiderivative is \(x^2-\frac{x^3}{3}\). Evaluating at the bounds gives
Washers: Use horizontal slices, perpendicular to the \(y\)-axis. The \(y\)-values run from \(0\) to \(2\). At height \(y\), the slanted boundary gives \(x=2-y\), while the other boundary is \(x=0\). Rotating this segment about the \(y\)-axis makes a disk of radius \(2-y\), with inner radius \(0\). Thus,
Check: Both methods give \(\frac{8\pi}{3}\) cubic units. In the shell setup, the variable describes the horizontal position; in the washer setup, it describes the vertical position. The different-looking integrands account for different slices through the same triangular solid.
Why the Two Integrals Must Agree
The preceding example illustrates a general principle. A shell integral and a washer integral do not add the same pieces in the same order. One sums thin cylindrical layers; the other sums thin cross-sectional disks or washers. But if both slice descriptions cover the same solid without gaps or overlap, each limiting sum measures its volume.
Partition the slice interval into thin pieces. On each piece, cross-sectional area times thickness approximates the volume of that short portion of the solid.
As the thicknesses approach zero, the sum of area-times-thickness approaches the washer integral, provided the boundary functions are integrable on the interval.
Partition by distance from the axis. Circumference times shell height times thickness approximates each thin layer's volume; the limiting sum is the shell integral.
Both limits are the volume of the same solid. Therefore, when both descriptions and their bounds are valid, the washer and shell integrals have equal values.
This reasoning justifies using agreement as a check. It does not guarantee that a particular setup is correct: two mistakes can sometimes produce matching answers. Verify each method from the region and axis before relying on the comparison.
Worked Example: The Same Triangle Rotated About the \(x\)-Axis
Worked Example: The Same Triangle Rotated About the \(x\)-Axis
Use the region bounded by \(x=0\), \(y=0\), and \(y=2-x\), but rotate it about the \(x\)-axis.
Washers: Vertical slices are perpendicular to the axis. For \(0\le x\le2\), the slice extends from \(y=0\) to \(y=2-x\). It creates a disk with outer radius \(2-x\) and inner radius \(0\). Therefore,
Shells: Horizontal slices are parallel to the \(x\)-axis. Their \(y\)-values run from \(0\) to \(2\). At height \(y\), the line \(y=2-x\) gives \(x=2-y\), so the segment runs from \(x=0\) to \(x=2-y\). Its length is \(2-y\), and its distance from the \(x\)-axis is \(y\). Thus,
Check: Both integrals again give \(\frac{8\pi}{3}\) cubic units. The axis changed, so the role of vertical and horizontal slices changed as well; the bounds and radii were determined anew rather than carried over from the first rotation.
Worked Example: A Curved Region Rotated About the \(y\)-Axis
Worked Example: A Curved Region Rotated About the \(y\)-Axis
Let the region lie under \(y=x^2\), above \(y=0\), and between \(x=0\) and \(x=2\). Rotate it about the \(y\)-axis.
Shells: A vertical slice at \(x\) has height \(x^2-0=x^2\) and radius \(x\). Its bounds are \(0\le x\le2\), so
Washers: Horizontal slices have \(0\le y\le4\), since the curve reaches \(y=4\) at \(x=2\). Solving \(y=x^2\) for the nonnegative \(x\)-coordinate gives \(x=\sqrt y\). At height \(y\), the region runs from \(x=\sqrt y\) to \(x=2\). On rotation about the \(y\)-axis, the outer radius is \(2\) and the inner radius is \(\sqrt y\). Therefore,
Check: Both setups give \(8\pi\) cubic units. At \(y=0\), the horizontal slice runs from \(x=0\) to \(x=2\); at \(y=4\), it has zero length at \(x=2\). These endpoint checks support the washer bounds and radii.
A Common Mistake: Reversing the Washer Radii
In the curved-region example, the washer's outer radius is \(2\), and its inner radius is \(\sqrt y\). A common error is to subtract the outer-radius square from the inner-radius square:
A negative volume signals that the radii have been reversed. The washer area is the area of the larger disk minus the area of the hole, so the correction is
Another warning sign is a mismatch between the slice direction and the variable of integration. Before evaluating, ask whether the bounds describe the full horizontal or vertical range required by that method. Then check a boundary slice: does its radius or shell height agree with the region?
Using Agreement as a Practical Check
A useful consistency check has three parts. First, sketch or describe the region and mark the axis of rotation. Second, build each setup independently: for shells, record radius, height, and bounds; for washers, record outer radius, inner radius, and bounds. Third, evaluate exactly where possible and compare.
If the answers disagree, inspect the geometry before the arithmetic. Common causes include using a height as a radius, omitting the inner radius of a washer, using the wrong inverse boundary for horizontal slices, or integrating over the wrong variable's interval. Check units too: a length multiplied by another length and integrated across a length should produce cubic units.
The main lesson is not that every volume should be computed twice. It is that, when both slice orientations give manageable integrals, they offer independent descriptions of one solid. Agreement strengthens confidence in the result; disagreement tells you to recheck the setups.
Check Your Understanding
For each question, identify the geometric information needed before evaluating an integral.
- For the region bounded by \(x=0\), \(y=0\), and \(y=3-x\), rotated about the \(y\)-axis, what are the shell radius, shell height, and shell bounds?
- For the same region rotated about the \(y\)-axis, what are the washer radii and the washer bounds?
- When a horizontal slice of a region is rotated about the \(y\)-axis, why is the shell radius measured horizontally rather than vertically?
- If a washer calculation gives a negative volume, which part of the setup should you check first?
- Why should matching shell and washer results be treated as a consistency check rather than as proof that both setups are correct?
