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Avoiding Double Counting When the Axis Crosses the Region

Use distance from the axis and the union of opposite-side slices to set up shell integrals without counting the same cylindrical layer twice.

Intermediate 10 min read

What You'll Learn

  • Explain why slices on opposite sides of an axis can generate the same cylindrical shell.
  • Express shell height as the length of a union of cross-sections at equal distances from the axis.
  • Decide when to restrict a shell integral to one side and when to combine both sides.
  • Split a radial integral at values where the contributing cross-sections or their overlap changes.
  • Identify and correct the overcounting caused by integrating across both sides with absolute-value radii.

Why Crossing the Axis Requires Care

A vertical slice at \(x=2\) and a vertical slice at \(x=-2\) are different slices in the original plane. But when both are rotated about the \(y\)-axis, each lies at distance \(2\) from that axis. They therefore sweep out the same cylindrical surface. If both slices contain the same heights, adding their shell contributions counts those heights twice.

The key is to describe a shell by its radius, which is a nonnegative distance from the axis, rather than by a signed \(x\)-coordinate. At each radius, collect the heights contributed by both sides and count the union of those heights only once. This also handles the case where the two sides contribute different, partly overlapping, or disjoint vertical intervals.

Key idea: When a region crosses the \(y\)-axis, the shell at radius \(r\) is generated by both \(x=r\) and \(x=-r\). Its height is the total length of the union of the vertical cross-sections at those two positions, not automatically the sum of their lengths.

Shell Height at a Nonnegative Radius

Consider a bounded planar region \(R\) rotated about the \(y\)-axis. For a fixed \(x\), let \(V(x)\) be the set of \(y\)-values for which \((x,y)\) lies in \(R\). A vertical cross-section may be one interval or several intervals. At radius \(r\geq 0\), the shell can be generated by points from either side of the axis, so define its vertical set by

$$ U(r)=V(r)\cup V(-r). $$

Write \(L(r)\) for the total length of the intervals in \(U(r)\), counting any overlap only once. The shell has radius \(r\), thickness \(dr\), and total height \(L(r)\). Thus, when the cross-sections and their boundaries are sufficiently well-behaved for the shell integral, the volume is

$$ V_{\text{solid}}=2\pi\int_0^{R_{\max}} rL(r)\,dr, $$

where \(R_{\max}\) is the greatest distance from the \(y\)-axis reached by the region. If \(V(r)\) and \(V(-r)\) are disjoint, their lengths add. If they overlap, the shared heights contribute only once. The same principle applies to rotation about the \(x\)-axis: use horizontal cross-sections at \(y=r\) and \(y=-r\), and take the union of their \(x\)-values.

Why the Union Formula Works

For a thin radial interval from \(r\) to \(r+\Delta r\), the points of the region at distances in that interval sweep out one or more thin cylindrical shells. At each height in \(U(r)\), the circumference is approximately \(2\pi r\), and the thickness is approximately \(\Delta r\). Heights shared by the two opposite-side cross-sections correspond to the same part of the rotated solid, so they appear once in \(U(r)\), not twice.

The thin-shell volume is therefore approximated by \(2\pi rL(r)\Delta r\). Adding these contributions over radial intervals gives Riemann sums for \(2\pi\int_0^{R_{\max}}rL(r)\,dr\). Taking the limit as the largest thickness tends to zero gives the formula. The argument depends on measuring the union of the cross-sections: replacing that union with the sum of their lengths would construct a sum of shell pieces, including pieces that may occupy the same part of space.

Practical rule: First replace the signed coordinate by distance \(r\geq 0\). Then find every cross-section at that distance, combine their vertical intervals as a union, and integrate the resulting total length.

Worked Example: A Rectangle That Crosses the \(y\)-Axis

Worked Example: A Rectangle That Crosses the \(y\)-Axis

Rotate the rectangle \(-2\leq x\leq 1\), \(0\leq y\leq 3\) about the \(y\)-axis. For \(0\leq r<1\), both \(x=r\) and \(x=-r\) lie in the rectangle. Each cross-section is the interval \([0,3]\), so their union has length \(3\). For \(1<r\leq 2\), only \(x=-r\) lies in the rectangle, and the union still has length \(3\). Thus \(L(r)=3\) for \(0\leq r\leq 2\). The value at the single boundary radius \(r=1\) does not affect the integral.

The shell integral is

$$ V_{\text{solid}} =2\pi\int_0^2 3r\,dr =6\pi\left[\frac{r^2}{2}\right]_0^2 =6\pi\left(\frac{4}{2}-0\right) =12\pi. $$

As a check, rotating the rectangle fills a cylinder of radius \(2\) and height \(3\), whose volume is \(\pi(2)^2(3)=12\pi\). The shell integral agrees. In contrast, integrating \(2\pi |x|(3)\) over \(-2\leq x\leq 1\) gives \(6\pi(2+\frac12)=15\pi\), which overcounts the shared shells for radii below \(1\).

When the Opposite-Side Cross-Sections Differ

The union is not always a single interval, and its length is not always just the larger of the two cross-section lengths. If the intervals overlap partly, count the shared part once. If they are separated, count the lengths of both intervals. The next examples illustrate these distinctions.

Worked Example: Disjoint Height Bands on Opposite Sides

Let the region consist of the two rectangles \(-2\leq x\leq 0\), \(0\leq y\leq 1\), and \(0\leq x\leq 2\), \(2\leq y\leq 3\). Rotate this region about the \(y\)-axis. For every \(0<r<2\), one side contributes the interval \([0,1]\) and the other contributes \([2,3]\). These intervals are disjoint, so \(U(r)=[0,1]\cup[2,3]\) and \(L(r)=1+1=2\).

Therefore,

$$ V_{\text{solid}} =2\pi\int_0^2 2r\,dr =4\pi\left[\frac{r^2}{2}\right]_0^2 =4\pi(2) =8\pi. $$

For a check, each rectangle rotates into a cylinder of radius \(2\) and height \(1\). Their vertical bands are separated, so their volumes add: \(\pi(2)^2(1)+\pi(2)^2(1)=4\pi+4\pi=8\pi\). The union-length calculation counts both bands without treating them as overlapping.

Worked Example: Partly Overlapping Height Bands

Let the region consist of \(-2\leq x\leq 0\), \(0\leq y\leq 2\), and \(0\leq x\leq 2\), \(1\leq y\leq 3\). Rotate it about the \(y\)-axis. At each radius \(0<r<2\), the cross-sections are \([0,2]\) and \([1,3]\). Their overlap is \([1,2]\), of length \(1\). Their union is \([0,3]\), of length \(3\); equivalently, its length is \(2+2-1=3\).

Consequently,

$$ V_{\text{solid}} =2\pi\int_0^2 3r\,dr =6\pi\left[\frac{r^2}{2}\right]_0^2 =6\pi(2) =12\pi. $$

For a geometric check, each rectangle generates a cylinder of radius \(2\) and height \(2\). The vertical bands \([0,2]\) and \([1,3]\) overlap on a band of height \(1\); their union is one cylinder-shaped solid with height \(3\). Its volume is \(\pi(2)^2(3)=12\pi\), as computed by shells. Adding the two cylinder volumes without subtracting their overlap would give \(8\pi+8\pi=16\pi\), an overcount.

Restricting or Splitting the Setup

Sometimes the union rule simplifies. If the region lies entirely on one side of the axis, each positive radius corresponds to at most one cross-section, and the usual shell setup on that side is enough. If the region crosses the axis but the two cross-sections at radius \(r\) have disjoint heights, their lengths can be added. If they overlap, the overlap must be removed—or, more directly, the length of the union must be found.

The function \(L(r)\) may change when a boundary is reached or when two intervals begin or cease to overlap. In that case, split the integral at those radii. A useful setup process is:

1
Use distance from the axis.
Set the shell radius to \(r\geq 0\), so that positions on opposite sides are represented by \(x=r\) and \(x=-r\) for rotation about the \(y\)-axis.
2
Find the cross-sections.
Determine the vertical intervals in the region at both positions, including any multiple components.
3
Form their union.
Count shared heights once, and add lengths only for portions that are disjoint.
4
Split where needed.
Choose radial intervals on which the cross-sections and their overlap have the same description.
5
Integrate and check.
Evaluate \(2\pi\int rL(r)\,dr\), then compare the result with the geometry or an independent volume calculation when available.

Common Pitfall: Using Absolute Values Without Removing Overlap

A common incorrect approach is to integrate over the full signed coordinate interval and replace \(x\) by \(|x|\) in the radius. That gives every point a nonnegative radius, but it does not make points at \(x=r\) and \(x=-r\) distinct shells. In the rectangle example, this approach produced \(15\pi\), although the solid is a cylinder with volume \(12\pi\). The error occurs because the shared shells at radii from \(0\) to \(1\) are included from both sides.

Restricting the calculation to one side without checking the region can also omit solid. For example, in the disjoint-band example, using only the right-hand rectangle would count the band from \(y=2\) to \(y=3\) and miss the band from \(y=0\) to \(y=1\). The correction is not always “use only one side”; it is “count the union at each radius.”

Takeaway: Absolute value gives the correct radius but does not resolve double counting. The shell height must describe all—and only—the heights in the rotated solid at that radius.

Check Your Understanding

For each question, focus on which heights are present at a fixed nonnegative radius.

  1. When rotating about the \(y\)-axis, which two vertical cross-sections can contribute to the shell at radius \(r>0\)?
  2. If those cross-sections are the intervals \([0,2]\) and \([1,4]\), what is the length of their union?
  3. If the cross-sections on opposite sides are disjoint, how are their lengths combined to obtain the shell height?
  4. Why can integrating \(2\pi |x|\) times the slice height over both positive and negative \(x\)-values count a shell twice?
  5. What should be checked to decide whether the radial integral needs to be split into pieces?

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