Why Slice Orientation Is the First Choice
A volume problem can often be set up in more than one way. The useful first question is not “Which formula do I remember?” but “What does a thin slice of the region become when it rotates?” A slice perpendicular to the axis sweeps out a disk or washer. A slice parallel to the axis sweeps out a cylindrical shell.
After identifying those possibilities, check how each slice describes the region. One choice may use the given boundary functions directly, while the other may require inverse functions or separate integrals where the region’s boundary changes. Choosing well can make the setup shorter and less error-prone.
Match the Slice to the Axis
For rotation about a horizontal line, vertical slices are perpendicular to the axis and horizontal slices are parallel to it. Thus vertical slices lead to disks or washers, while horizontal slices lead to shells. For rotation about a vertical line, the roles reverse: horizontal slices lead to disks or washers, and vertical slices lead to shells.
| Axis of rotation | Perpendicular slices | Parallel slices |
|---|---|---|
| Horizontal | Vertical; disks or washers, usually \(dx\) | Horizontal; shells, usually \(dy\) |
| Vertical | Horizontal; disks or washers, usually \(dy\) | Vertical; shells, usually \(dx\) |
“Usually” matters: the variable of integration follows the direction of the slice. A vertical slice has thickness \(dx\), and a horizontal slice has thickness \(dy\). The axis may be shifted from the coordinate axes; the same orientation rule applies.
For a disk or washer, a cross-section has area \(\pi R^2\) or \(\pi(R^2-r^2)\). For a shell, a thin slice at radius \(r\), height \(h\), and thickness \(dt\) contributes approximately circumference times height times thickness, \(2\pi rh\,dt\). These formulas describe different slice orientations, so using the wrong one changes the geometry, not merely the notation.
A Reliable Selection Process
Decide whether it is horizontal or vertical, including if it is shifted.
Perpendicular slices suggest disks or washers; parallel slices suggest shells.
For washers, identify outer and inner radii. For shells, identify radius and height.
Ask whether each radius or height has the same boundary expression throughout the interval. If not, determine where it changes and split the integral.
Prefer a setup with direct boundary functions, fewer splits, and manageable integration.
This is a decision method, not a rule that one technique is always superior. A setup with no split can still be harder to integrate, and a split integral can still be preferable if its expressions are simpler.
Why the Orientation Rule Works
Consider a small slice of a planar region. If it is perpendicular to the axis, its distance from the axis changes across the slice, producing a circular cross-section. If the slice reaches the axis, the cross-section is a disk; if it stays away from the axis, the rotation leaves a hole and produces a washer. Its volume is cross-sectional area times a small thickness along the axis.
If the slice is parallel to the axis, it lies at a particular distance from the axis. Rotating it makes a thin cylindrical shell. Its circumference is \(2\pi r\), and its height is the slice length \(h\). Multiplying circumference by height and by the small radial thickness gives the shell volume element. This explains why the slice’s orientation determines both the formula and the integration variable.
Worked Example: Shells Avoid a Washer Split
Worked Example: Shells Avoid a Washer Split
Let the region be bounded by \(y=x\), \(y=2-x\), and the \(x\)-axis. Rotate it about the \(x\)-axis. Decide which method gives the cleaner setup, then find the volume.
The two sloping boundaries meet when \(x=2-x\), so \(2x=2\) and \(x=1\). The region has vertices \((0,0)\), \((1,1)\), and \((2,0)\). Since the axis is horizontal, horizontal slices are parallel to it and form shells. At height \(y\), the left endpoint is \(x=y\) and the right endpoint is \(x=2-y\), for \(0\le y\le1\). Thus the shell height is \((2-y)-y=2-2y\), and its radius is \(y\).
Vertical slices are perpendicular to the axis, so washers are also possible. However, the top boundary changes at \(x=1\): it is \(y=x\) on \([0,1]\) and \(y=2-x\) on \([1,2]\). The washer integral must therefore split. The horizontal-shell setup has one interval and one expression, so use shells:
An antiderivative of \(2y-2y^2\) is \(y^2-\frac{2}{3}y^3\). Therefore,
Check the integration: \(\frac{d}{dy}(y^2-\frac{2}{3}y^3)=2y-2y^2\), the integrand. The antiderivative is \(0\) at \(y=0\) and \(1/3\) at \(y=1\), so the integral is \(1/3\), giving \(V=2\pi/3\).
Worked Example: Washers Avoid a Shell Split
Worked Example: Washers Avoid a Shell Split
Use the same triangular region, but now rotate it about the \(y\)-axis. Choose the cleaner method and compute the volume.
The axis is vertical. Horizontal slices are perpendicular to it, so they form washers. At a height \(y\) from \(0\) to \(1\), the slice runs from \(x=y\) to \(x=2-y\). Rotating about the \(y\)-axis gives inner radius \(r=y\) and outer radius \(R=2-y\). These expressions hold throughout the interval; no split is needed.
Expand each square: \((2-y)^2=4-4y+y^2\), so the difference is \(4-4y\). Hence
Check the antiderivative: \(\frac{d}{dy}(4y-2y^2)=4-4y\). Its endpoint values are \(2\) at \(y=1\) and \(0\) at \(y=0\), confirming the integral and the volume.
For comparison, vertical slices are parallel to the \(y\)-axis and produce shells. Their height changes at \(x=1\): it is \(x\) from \(0\) to \(1\), then \(2-x\) from \(1\) to \(2\). That shell setup needs two integrals. Here washers are the cleaner choice.
Worked Example: Compare the Variables Before Integrating
Worked Example: Compare the Variables Before Integrating
Take the region bounded by \(y=x^2\), the \(x\)-axis, and \(x=2\), and rotate it about the \(x\)-axis. Compare the two orientations and use the more direct setup to find the volume.
The axis is horizontal. Vertical slices are perpendicular to it, so they form disks. For \(0\le x\le2\), each disk has radius \(x^2\); there is no inner radius because the region reaches the axis. This gives a polynomial integral.
Check: \(\frac{d}{dx}(x^5/5)=x^4\), and the antiderivative takes values \(32/5\) and \(0\) at \(2\) and \(0\), respectively.
Horizontal slices are parallel to the axis and would make shells. To express their left endpoint using \(y\), solve \(y=x^2\) for \(x=\sqrt y\). For \(0\le y\le4\), the shell height is \(2-\sqrt y\) and its radius is \(y\). This is a valid, unsplit setup, but its integrand contains a square root. The disk method uses the given boundary directly and is simpler to integrate.
Common Mistake: Using the Shell Formula on a Perpendicular Slice
In the first triangular-region example, a tempting but incorrect setup is \(2\pi\int_0^2 x(\text{top}-\text{bottom})\,dx\). It treats vertical slices as shells about the \(x\)-axis and uses \(x\) as the radius. But vertical slices are perpendicular to that horizontal axis, so they form washers, not shells. Also, \(x\) measures horizontal distance, not distance from the \(x\)-axis; that radial distance is vertical.
Correct the method by using horizontal slices for shells: their radius is \(y\), their height is \(2-2y\), and \(0\le y\le1\). Alternatively, use vertical slices as washers and split at \(x=1\), where the top boundary changes. In either method, identify the orientation and the actual geometric dimensions before writing an integral.
Choosing Efficiently
When both methods are possible, compare more than the number of integrals. Check whether you can describe the slice using the original functions or must solve for inverse functions; whether the outer and inner radii remain in the same order; and whether the shell height or washer radii change expression within the interval. A split is necessary whenever the geometric description changes, but it is not a problem if the resulting integrals are clear.
Keep the distinction between orientation and convenience precise. Orientation determines which formula applies to a proposed slice. Convenience helps decide which valid orientation to use. A direct formula is not automatically valid unless its radius, height, bounds, and thickness all match the geometry.
Check Your Understanding
For each question, identify the slice orientation and explain what makes a method convenient or inconvenient.
- For rotation about a vertical axis, which slice direction produces washers, and which produces shells?
- A region rotated about the \(x\)-axis has a top boundary that changes at \(x=3\). What should you check before using vertical washers?
- For the region under \(y=x^2\) from \(x=0\) to \(x=2\), what are the radius and height of a horizontal shell about the \(x\)-axis?
- Why might an unsplit integral still be less convenient than a split integral?
- When rotating about the line \(y=4\), what determines a shell’s radius: its \(x\)-coordinate or its distance from \(y=4\)?
