Why the Axis Position Matters
For a cylindrical-shell setup, the radius is the distance from a thin slice of the region to the axis of rotation. When the axis is the \(y\)-axis or the \(x\)-axis, this distance is often easy to recognize. If the axis is shifted, the same idea applies, but the radius must be measured from the shifted line, not from the origin.
There is an important sign issue. A distance cannot be negative. The expression for the radius therefore depends on which side of the axis contains the region. If you use the wrong signed expression, the resulting integral may give a negative value, which cannot represent a volume.
Radius for a Shifted Axis
A vertical line has equation \(x=c\). A vertical shell at coordinate \(x\) has radius equal to the horizontal distance from \(x\) to \(c\). A horizontal line has equation \(y=d\). A horizontal shell at coordinate \(y\) has radius equal to the vertical distance from \(y\) to \(d\).
In particular, if every point of the region has \(x\ge c\), then \(r(x)=x-c\). If every point has \(x\le c\), then \(r(x)=c-x\). For a horizontal axis, if every point has \(y\ge d\), then \(r(y)=y-d\); if every point has \(y\le d\), then \(r(y)=d-y\).
For a vertical shell, the slice is parallel to the vertical axis, so the shell height is the length of the vertical segment through the region. For a horizontal shell, the height is the length of the horizontal segment. Once radius \(r\) and height \(h\) are known, the shell method uses the integral \(V=2\pi\int rh\,d(\text{coordinate})\), with respect to \(x\) for vertical shells and \(y\) for horizontal shells.
Why the Shell Formula Uses This Radius
Consider a thin slice parallel to the axis of rotation, with radius \(r\), height \(h\), and thickness \(\Delta\). When revolved, it forms a thin cylindrical shell. Its circumference is \(2\pi r\). For a sufficiently thin shell, its volume is approximately its circumference multiplied by its height and thickness:
Adding the volumes of the thin shells across the region gives a sum of terms \(2\pi rh\Delta\). As the thicknesses shrink toward zero, these sums become the definite integral of \(2\pi rh\). The radius in this derivation is a distance from the axis, which explains why it must be nonnegative.
Worked Examples with a Vertical Axis
Worked Example: A Region to the Right of a Shifted Vertical Axis
Consider the region bounded by \(x=0\), \(x=2\), \(y=0\), and \(y=4-x\). Rotate it about the vertical line \(x=-2\). Set up and evaluate the shell integral.
Step 1: Check which side of the axis contains the region. The region has \(0\le x\le 2\), while the axis is at \(x=-2\). Thus every point is to the right of the axis, so the radius is \(x-(-2)=x+2\). On the interval, \(2\le r(x)\le4\), so the radius is nonnegative throughout.
Step 2: Find the shell height. A vertical slice extends from \(y=0\) to \(y=4-x\), so \(h(x)=(4-x)-0=4-x\). The bounds are \(0\le x\le2\).
Step 3: Form and evaluate the integral.
Check. The derivative of the antiderivative \(-x^3/3+x^2+8x\) is \(-x^2+2x+8=(x+2)(4-x)\), as required. Both radius and height are positive throughout the interval, so the positive volume is consistent with the geometry.
Worked Examples with a Horizontal Axis
Worked Example: A Rectangle Above a Shifted Horizontal Axis
Let the region be the rectangle \(1\le x\le4\), \(0\le y\le2\). Rotate it about \(y=-3\), using horizontal shells.
Step 1: Determine the radius. Every point of the rectangle is above the axis, since \(y\ge0>-3\). A shell at height \(y\) has radius \(y-(-3)=y+3\). For \(0\le y\le2\), the radius ranges from \(3\) to \(5\), so it is nonnegative everywhere.
Step 2: Determine the height. Each horizontal slice runs from \(x=1\) to \(x=4\). Its length is \(h(y)=4-1=3\). The \(y\)-bounds are \(0\) and \(2\).
Step 3: Integrate.
Check. The derivative of \(6\pi(y^2/2+3y)\) is \(6\pi(y+3)\), the integrand. The height is \(3\), and the radius stays between \(3\) and \(5\), so every shell contributes positive volume.
Worked Example: A Triangular Region Below a Shifted Horizontal Axis
Consider the triangle bounded by \(x=0\), \(y=x\), and \(y=2\). Rotate it about the horizontal line \(y=5\), using horizontal shells.
Step 1: Express the horizontal slice. The \(y\)-values run from \(0\) to \(2\). At a fixed \(y\) in this range, the slice runs from \(x=0\) to \(x=y\), so \(h(y)=y-0=y\).
Step 2: Find and check the radius. The region is below \(y=5\), so the distance to the axis is \(r(y)=5-y\), not \(y-5\). For \(0\le y\le2\), \(3\le r(y)\le5\); thus the radius is nonnegative throughout.
Step 3: Evaluate the shell integral.
Check. Differentiating \(5y^2/2-y^3/3\) gives \(5y-y^2=(5-y)y\), the integrand. The shell height is nonnegative on the interval and the radius is at least \(3\), consistent with a positive volume.
A Sign Mistake That Produces a Negative Volume
Suppose a rectangular region has \(-4\le x\le-3\) and height \(1\), and is rotated about \(x=-2\). The region lies to the left of the axis. Its radius is therefore \((-2)-x=-2-x\). The expression \(x-(-2)=x+2\) has the wrong sign on this interval.
The correct setup gives:
The incorrect calculation would use \(x+2\) as the radius:
The negative result signals a setup error, not a negative physical volume. On \([-4,-3]\), \(x+2\) ranges from \(-2\) to \(-1\); the actual distance is its opposite, \(-(x+2)=-2-x\).
A Reliable Radius Check
Before integrating, identify the axis and the coordinate values occupied by the whole region. For a vertical axis \(x=c\), check the region's \(x\)-coordinates; for a horizontal axis \(y=d\), check its \(y\)-coordinates. Then test the proposed radius at the interval endpoints, or otherwise establish its sign throughout the interval.
Write it as \(x=c\) for a vertical axis or \(y=d\) for a horizontal axis.
Determine whether all its relevant coordinates are greater than or equal to the axis coordinate, or less than or equal to it.
Choose \(x-c\) or \(c-x\), or \(y-d\) or \(d-y\), so the radius is nonnegative throughout.
Check the radius at the bounds and confirm the shell height is the slice length. Then substitute both into the shell formula.
This check is especially useful when the axis is shifted to a negative coordinate or lies on the opposite side of the region from the origin. The origin does not determine the radius; the distance to the axis does. If the region is not wholly on one side of the axis, the signed difference may not stay nonnegative, so this single-side setup does not apply unchanged.
Check Your Understanding
For each question, identify the correct radius and explain how you know it is nonnegative on the stated interval.
- A region has \(2\le x\le5\) and is rotated about \(x=-1\) using vertical shells. What is the shell radius?
- A region has \(-3\le x\le-1\) and is rotated about \(x=2\) using vertical shells. What is the shell radius?
- A region has \(1\le y\le4\) and is rotated about \(y=6\) using horizontal shells. What is the shell radius?
- A region has \(-2\le y\le1\) and is rotated about \(y=-5\) using horizontal shells. What is the shell radius?
- Why does a negative value for a proposed shell radius indicate that the radius expression needs to be checked?
