Reading Radius, Height, and Thickness from a Slice

Reading a Slice Before Setting Up a Shell

A shell setup begins with a geometric question: what does one thin slice of the region become when it is rotated? The answer depends on how the slice is oriented relative to the axis of rotation. A slice parallel to the axis sweeps out a thin cylindrical shell. Its distance to the axis gives the shell radius; the length of the slice within the region gives the shell height; and the small movement between neighboring slices gives the thickness.

These measurements are easy to confuse when reading a sketch. The radius is not the slice’s length, and the thickness is not the distance from the slice to the axis. A reliable reading starts by identifying the axis, then the slice orientation, and only then the three dimensions.

Which Slices Produce Shells?

A vertical slice is a narrow strip whose long direction is vertical. Its position changes horizontally, so its thickness is \(dx\). A horizontal slice is a narrow strip whose long direction is horizontal. Its position changes vertically, so its thickness is \(dy\).

For the shell method, the slice must be parallel to the axis of rotation. Thus a vertical axis pairs with vertical slices, while a horizontal axis pairs with horizontal slices. A slice perpendicular to the axis instead sweeps out a disk or washer-like cross-section; it is not the shell slice for that axis.

This is an orientation rule, not a rule based on whether the graph is “above” or “below” something. Look at the direction of the slice and compare it with the direction of the axis.

Radius Is a Perpendicular Distance

The shell radius is measured from the slice to the axis along a line perpendicular to that axis. For a vertical axis \(x=c\), a vertical slice at \(x\) has radius \(|x-c|\). For a horizontal axis \(y=d\), a horizontal slice at \(y\) has radius \(|y-d|\). The absolute value expresses that distance is nonnegative.

When the region lies entirely on one side of the axis, the absolute value can be written more simply. For example, if \(x>c\), then \(|x-c|=x-c\). If \(y<d\), then \(|y-d|=d-y\). Do not choose the subtraction order by memorizing a formula: identify which coordinate is farther from the axis and compute the distance.

Height Is the Slice’s Length Inside the Region

Once the slice is parallel to the axis, its height is found by locating where that slice enters and leaves the region. For a vertical slice, this means reading the lower and upper \(y\)-coordinates. If those boundaries are \(y_{\text{lower}}(x)\) and \(y_{\text{upper}}(x)\), then the slice length is \(y_{\text{upper}}(x)-y_{\text{lower}}(x)\). For a horizontal slice, read the left and right \(x\)-coordinates; the length is \(x_{\text{right}}(y)-x_{\text{left}}(y)\).

The labels “upper,” “lower,” “left,” and “right” describe the slice’s endpoints within the region. They do not describe the axis. A quick sketch of the slice, with both endpoints marked, can prevent a sign error in the height.

From Dimensions to the Shell Differential

The shell formula combines the three measurements read from the slice. A shell with radius \(r\), height \(h\), and small thickness \(\Delta s\) has volume approximately equal to its circumference times its height times its thickness. Here \(\Delta s\) is \(dx\) for vertical slices and \(dy\) for horizontal slices.

The differential form is the one used in a shell integral. The approximation for a finite thickness has a small correction, so it should not be treated as an exact finite-shell formula. For instance, if a shell of height \(h\) extends from radius \(r\) to radius \(r+\Delta r\), its exact volume is

For fixed \(r>0\) and \(h>0\), the final term is smaller than the first-order term when \(\Delta r\) is sufficiently small: their ratio is \(\Delta r/(2r)\), which tends to zero as \(\Delta r\) tends to zero. At \(r=0\), the first-order term vanishes, so it is not correct to describe the correction as smaller than that term. The differential formula still holds in the limiting sense: at \(r=0\), the exact volume divided by \(\Delta r\) is \(\pi h\Delta r\), which tends to \(0=2\pi rh\).

Worked Examples

A Reliable Reading Procedure

When a sketch is given, use the following sequence before writing a volume integral. Each step answers a separate question, so keeping them separate makes errors easier to spot.

Common Mistake: Using the Wrong Differential

In the triangular-region example, a common error is to use \(dx\) because the boundary includes the expression \(x=2y\), even though the chosen slice is horizontal. That would give the incorrect shell contribution \(2\pi(3)(2)\,dx=12\pi\,dx\). The boundary equation does not determine the thickness; the slice orientation does. Since neighboring horizontal slices move vertically, the correct contribution uses \(dy\): \(2\pi(3)(2)\,dy=12\pi\,dy\).

Another frequent mix-up is to call the slice length the radius. A radius reaches from the slice to the axis; a height stays within the region along the slice. If the axis is drawn and the slice endpoints are marked, these two measurements have different visible roles.

Why Careful Slice Reading Matters

A shell integral records the volume generated by many thin slices. If the radius is wrong, the circumference is wrong; if the height is wrong, the amount of region represented by each shell is wrong; and if the differential is wrong, the slices are accumulated in the wrong direction. All three choices come from the geometry before any integration begins.

For a simple region lying entirely on one side of the axis, the distance formulas above directly identify the radius. If the axis passes through the region, a slice’s distance may not describe a single non-overlapping shell contribution. That situation requires additional care rather than blindly applying the same reading to the entire region.