Why a Shifted Axis Changes the Radius
The shell method builds a solid from thin slices parallel to the axis of rotation. When the axis is shifted away from the coordinate axis, the shell's radius is no longer simply \(x\) or \(y\). It is the distance from the slice to the actual line of rotation.
For a vertical axis \(x=a\), use vertical slices. Each slice rotates into a cylindrical shell whose radius is the horizontal distance from \(x\) to \(a\). For a horizontal axis \(y=b\), use horizontal slices, and the radius is the vertical distance from \(y\) to \(b\).
Shell Formulas for Shifted Lines
Suppose a region lies between \(x=p\) and \(x=q\), on one side of the vertical line \(x=a\). A vertical slice at \(x\) has height \(h(x)\) and thickness \(dx\). Its shell radius is the distance \(|x-a|\). The shell method gives:
This form applies directly when every vertical slice in the interval lies on the same side of \(x=a\). On the side \(x<a\), the radius is \(a-x\); on the side \(x>a\), it is \(x-a\). In either case, the radius is positive or zero.
Similarly, suppose a region lies between \(y=r\) and \(y=s\), on one side of the horizontal line \(y=b\). A horizontal slice at \(y\) has length \(w(y)\) and thickness \(dy\). Its shell radius is \(|y-b|\), so:
These formulas match slice direction to the axis: vertical slices for a vertical axis, and horizontal slices for a horizontal axis. The slice direction is parallel to the axis, so its rotation creates a shell. The radius is measured perpendicular to the axis.
Why the Shell Formula Works
A thin slice parallel to the axis, when rotated, forms a thin cylindrical shell. If its radius is \(R\), height is \(H\), and thickness is \(\Delta\), its volume is approximately circumference times height times thickness:
For a vertical axis, the slice thickness is \(\Delta x\), the radius is the distance from \(x\) to \(a\), and the height is \(h(x)\). Adding the volumes of many thin shells gives a sum that approaches an integral as the thicknesses shrink. This gives the vertical-shell formula. For a horizontal axis, the same reasoning uses thickness \(\Delta y\), radius \(|y-b|\), and slice length \(w(y)\).
Worked Examples
Worked Example: A Rectangle Rotated About a Shifted Vertical Line
The rectangle is \(0\le x\le2\), \(0\le y\le3\). Rotate it about \(x=4\). Find the volume using shells.
The axis is to the right of the entire rectangle, so vertical slices produce shells counted once. The shell height is \(3-0=3\), and the radius at \(x\) is \(4-x\). The bounds are \(x=0\) and \(x=2\):
The antiderivative checks because \(\frac{d}{dx}(4x-x^2/2)=4-x\). As a geometric check, the rotated rectangle forms a hollow cylinder of height \(3\), outer radius \(4\), and inner radius \(2\). Its volume is \(\pi(4^2-2^2)(3)=\pi(16-4)(3)=36\pi\), agreeing with the shell integral.
Worked Example: A Region Under a Curve About a Shifted Vertical Line
The region is bounded by \(y=x^2\), \(y=0\), \(x=0\), and \(x=2\). Rotate it about \(x=-1\). Set up and evaluate the shell integral.
The region lies to the right of \(x=-1\). A vertical slice at \(x\) has height \(x^2-0=x^2\), and its radius is \(x-(-1)=x+1\). Thus:
The antiderivative checks because \(\frac{d}{dx}(x^4/4+x^3/3)=x^3+x^2\). The radius factor is \(x+1\), not \(x\), because the axis is one unit to the left of the \(y\)-axis.
Worked Example: A Rectangle Rotated About a Shifted Horizontal Line
The rectangle is \(0\le x\le2\), \(1\le y\le3\). Rotate it about \(y=-1\). Find the volume using horizontal shells.
The axis lies below the entire rectangle. A horizontal slice has length \(2-0=2\), and its radius is \(y-(-1)=y+1\). The bounds are \(y=1\) and \(y=3\):
The antiderivative checks because \(\frac{d}{dy}(y^2/2+y)=y+1\). Independently, the solid is a hollow cylinder with length \(2\), outer radius \(4\), and inner radius \(2\). Its volume is \(\pi(4^2-2^2)(2)=24\pi\).
Worked Example: Horizontal Shells with a Variable Slice Length
The region is bounded by \(x=0\), \(x=y\), \(y=1\), and \(y=3\). Rotate it about \(y=-1\). Set up and evaluate the shell integral.
The region lies above the axis. At a given \(y\), the horizontal slice runs from \(x=0\) to \(x=y\), so its length is \(w(y)=y\). Its shell radius is \(y+1\). Therefore:
The antiderivative checks because \(\frac{d}{dy}(y^3/3+y^2/2)=y^2+y\). The changing slice length \(y\) belongs in the height factor; the shifted-axis distance \(y+1\) belongs in the radius factor.
A Crucial Caution: Do Not Count the Same Radius Twice
The distance expressions \(|x-a|\) and \(|y-b|\) correctly measure radii, but that does not mean they can always be integrated over a region that crosses the axis. Slices on opposite sides of an axis can rotate into the same shells. Summing both sets of slice volumes would count some of the solid more than once.
For example, take the rectangle \(-1\le x\le1\), \(0\le y\le1\), and rotate it about \(x=0\). The solid is a cylinder of radius \(1\) and height \(1\), so its volume is \(\pi\). The tempting calculation
is wrong: it counts the shells generated by the left half and the right half separately, even though those halves produce the same radii. Since this particular region is symmetric, using just the right half counts every shell once:
For a region that crosses an axis, do not apply the one-side formula over all slices without checking whether shells repeat. The calculation must be split or reformulated so each resulting shell is counted once. The absolute value supplies a nonnegative distance; it does not prevent double-counting.
A Reliable Setup Routine
Record its equation as \(x=a\) or \(y=b\); the constant determines the distance measurement.
Use vertical slices for \(x=a\), and horizontal slices for \(y=b\).
The radius is a distance to the axis; the height or width is the length of the slice in the region.
If all slices lie on one side, the direct formula applies. If the region crosses the axis, check for repeated radii before integrating.
Use \(dx\) with vertical slices and \(dy\) with horizontal slices, then evaluate the integral.
Check Your Understanding
For each setup, identify the shell radius and slice length before evaluating any integral.
- A region lies between \(x=1\) and \(x=4\), with vertical slice height \(h(x)\), and is rotated about \(x=6\). What is the shell radius at \(x\), and what differential belongs in the integral?
- A region lies between \(y=2\) and \(y=5\), with horizontal slice length \(w(y)\), and is rotated about \(y=-1\). State the shell radius and the integral setup.
- Why does the direct integral \(2\pi\int_p^q |x-a|h(x)\,dx\) require a check that the region lies on one side of \(x=a\)?
- For the rectangle \(0\le x\le3\), \(0\le y\le2\), rotated about \(x=5\), write the shell integral without evaluating it.
- What geometric error occurs if slices from both sides of an axis generate the same radii but are all included in a shell integral?
