Why Radius and Height Must Be Identified Separately
A cylindrical-shell setup uses two lengths that are easy to confuse. The radius tells how far a representative slice is from the axis of rotation. The height tells how long that slice is. Both are lengths, but they come from different geometric measurements.
That distinction matters even though the shell formula multiplies the two lengths. If you merely exchange the names of two correct factors, their product does not change. But if you mistake a coordinate for a distance, or a curve value for the length between two boundaries, the resulting shell may not describe the region at all.
Definitions from a Representative Slice
Imagine a thin slice of a planar region, positioned so that rotating it about an axis produces a cylindrical shell. The slice is parallel to the axis of rotation. Its thickness is measured perpendicular to the slice, and its length becomes the shell’s height.
For vertical slices rotated about a vertical line \(x=a\), the radius is the horizontal distance from the slice at \(x\) to that line: \(|x-a|\). The height is the vertical distance between the top and bottom boundaries of the region: \(y_{\text{top}}-y_{\text{bottom}}\).
For horizontal slices rotated about a horizontal line \(y=b\), the radius is the vertical distance from the slice at \(y\) to the axis: \(|y-b|\). The height is the horizontal distance between the right and left boundaries: \(x_{\text{right}}-x_{\text{left}}\).
The absolute value in the radius is important: distances are nonnegative. If the axis crosses the region, the distance expression can change form, and the region may need to be treated in separate parts so that each shell is counted correctly.
Why the Shell Formula Uses Both Lengths
A representative shell has circumference approximately \(2\pi r\), height \(h\), and thickness \(d\ell\). Multiplying circumference by height gives the approximate lateral area of the shell; multiplying by thickness gives its approximate volume. The resulting expression is \(2\pi rh\,d\ell\).
The same formula follows by considering a thin cylindrical layer of constant height \(h\), with inner radius \(r\) and outer radius \(r+\Delta r\). Its volume is the difference between the volumes of two cylinders:
After dividing by \(\Delta r\), the second term contributes \(\pi h\Delta r\), which approaches zero as the layer becomes thinner. Thus the volume per unit thickness approaches \(2\pi rh\). For a region whose shell height changes, the definite integral adds these thin-shell contributions across the interval.
Worked Example: A Shifted Axis Changes the Radius
Worked Example: A Shifted Axis Changes the Radius
The region lies between \(y=0\) and \(y=2-x\), for \(0\leq x\leq2\). Rotate it about the vertical line \(x=-1\). Find the volume using vertical shells.
Step 1: Identify a representative slice. A vertical slice at \(x\) runs from \(y=0\) to \(y=2-x\). Its length is the upper value minus the lower value:
Step 2: Find the radius from the axis. The slice is at \(x\), while the axis is at \(x=-1\). Their horizontal distance is
since \(0\leq x\leq2\). The radius is not \(x\): \(x\) measures distance from the \(y\)-axis, not from the given axis \(x=-1\).
Step 3: Set up and evaluate the integral.
Check. Differentiating the antiderivative gives \(-x^2+x+2=(x+1)(2-x)\), the integrand used. At \(x=1\), the shell has radius \(2\) and height \(1\), consistent with its distance from \(x=-1\) and its slice from \(y=0\) to \(y=1\).
Worked Example: A Horizontal Slice Has Its Own Length
Worked Example: A Horizontal Slice Has Its Own Length
Consider the region bounded by \(x=0\), \(x=y\), \(y=1\), and \(y=3\). Rotate it about \(y=-1\). Use horizontal shells.
Step 1: Determine the slice length. At height \(y\), the slice extends from \(x=0\) to \(x=y\). Its horizontal length is
Step 2: Determine the radius. The axis is \(y=-1\), so the vertical distance from the slice at \(y\) to the axis is
because \(1\leq y\leq3\). The shell height is a horizontal length; it is not the distance to the horizontal axis.
Step 3: Calculate the volume.
Check. The derivative of \(y^3/3+y^2/2\) is \(y^2+y\), which equals \((y+1)y\). At \(y=2\), the slice runs from \(x=0\) to \(x=2\), so its height is \(2\); the distance from \(y=2\) to the axis \(y=-1\) is \(3\). These are different geometric measurements.
Worked Example: Height Is the Difference Between Boundaries
Worked Example: Height Is the Difference Between Boundaries
The region is between \(y=2x\) and \(y=x^2\), for \(0\leq x\leq2\). Rotate it about the \(y\)-axis. Find its volume using vertical shells.
Step 1: Find which boundary is above. On this interval, \(2x-x^2=x(2-x)\geq0\), so \(y=2x\) is above \(y=x^2\). The shell height is the distance between them:
Step 2: Find the radius. The axis is \(x=0\), so for \(x\geq0\),
Step 3: Integrate and check.
Differentiating \(2x^3/3-x^4/4\) gives \(2x^2-x^3=x(2x-x^2)\), as required. At \(x=1\), the boundaries are \(y=2\) and \(y=1\), so the slice height is \(1\), not \(2\). Its radius is also \(1\), but for a separate reason: its distance from the \(y\)-axis is \(1\).
A Common Mistake: Using a Boundary Value as the Height
In the last example, a tempting but incorrect setup is to use \(2x\) as the shell height because \(2x\) is the top boundary. That value measures the vertical position of the top boundary above the \(x\)-axis; it does not measure the length of the slice between the two curves.
The incorrect integrand omits the part of the slice below \(y=2x\) and above \(y=x^2\). The correction is to subtract the lower boundary from the upper boundary before integrating:
The size of the discrepancy is a warning, but the geometric definition identifies the error: shell height is a length between the slice’s endpoints, not necessarily the value of either endpoint.
Swapping Factors Versus Misidentifying Them
There is an important subtlety. If the radius and height have both been found correctly, writing \(h\) before \(r\) in the product does not change the volume integrand, because \(rh=hr\). For example, the correct product in the first example is \((x+1)(2-x)\); writing \((2-x)(x+1)\) gives exactly the same integral.
That algebraic fact does not make radius and height interchangeable as geometric quantities. The radius determines the shell’s circumference, while the height is the length of the generating slice. Their separate meanings tell you which distance to calculate and help you detect errors such as using \(x\) instead of \(x+1\) for an axis at \(x=-1\), or using a curve value instead of the difference between two boundaries.
A Reliable Setup Routine
Record the slice’s position and the equation of the rotation axis.
Find the perpendicular distance from the slice to the axis, using a nonnegative distance.
Subtract the slice’s endpoint coordinates in the direction parallel to the axis.
Choose a convenient slice and confirm that the radius and height match the geometry.
Use \(2\pi rh\) times the slice thickness, and split the interval if a formula changes.
Check Your Understanding
For each question, identify the two lengths separately before forming a shell integral.
- Vertical slices at \(x\) are rotated about \(x=3\). What is the radius for a slice with \(x\leq3\)?
- A horizontal slice at \(y\) runs from \(x=1\) to \(x=4-y\). What is its height?
- For the region between \(y=5\) and \(y=x^2\), where \(0\leq x\leq2\), what is the vertical-shell height?
- Why does exchanging correctly identified \(r\) and \(h\) in the product \(2\pi rh\) leave the integral unchanged?
- For a vertical shell at \(x\) rotated about \(x=-2\), what is its radius when \(x\geq0\)?
