Horizontal Shells About the X-Axis

Why Use Horizontal Shells?

When a region rotates around the x-axis, a horizontal slice is parallel to the axis of rotation. As the slice revolves, it sweeps out a thin cylindrical shell. Adding the volumes of these shells gives a volume integral.

This method can be useful when the region’s left and right boundaries are naturally written as functions of \(y\). Instead of rearranging those boundaries to express \(y\) as a function of \(x\), we can measure each horizontal width directly and integrate with respect to \(y\).

Throughout this tutorial, assume the region lies on or above the x-axis and each horizontal slice is one interval. These conditions make the shell radius simply \(y\). More complicated situations, such as an axis that cuts through the region or boundaries that change, require additional care.

Radius, Height, and Thickness

Consider a horizontal slice at height \(y\), extending from \(x=x_{\text{left}}(y)\) to \(x=x_{\text{right}}(y)\). Rotating this slice about the x-axis forms a shell.

• The shell’s radius is the distance from the slice to the x-axis: \(y\).
• The shell’s height is the horizontal length of the slice: \(x_{\text{right}}(y)-x_{\text{left}}(y)\).
• The shell’s thickness is a small change in height, written \(dy\).

The word “height” here means the shell’s length along the x-direction. It is not the distance from the shell to the axis; that distance is its radius.

The bounds \(a\) and \(b\) are the lowest and highest y-values in the region. The integrand is \(2\pi\) times radius times height, so the integral adds the volumes of the shells across the region.

Why the Formula Works

A cylinder with radius \(r\), height \(h\), and thickness \(\Delta y\) has volume approximately equal to its circumference times its height times its thickness. For a thin shell at height \(y\), the circumference is approximately \(2\pi y\), and the shell’s length is the horizontal width of the region.

To justify adding these contributions, divide the interval from \(a\) to \(b\) into thin subintervals. In each subinterval, choose a representative height \(y_i\). The corresponding shell has approximate volume \(2\pi y_i\bigl(x_{\text{right}}(y_i)-x_{\text{left}}(y_i)\bigr)\Delta y\). Summing these approximations gives a Riemann sum. As the subinterval thicknesses approach zero, the sum approaches the definite integral:

The limit is the volume because the shells fill the solid, while the approximation error from treating each thin slice as a shell vanishes as the slices become thinner. The formula requires the horizontal width and radius to describe the shells across the stated y-interval; if the region’s horizontal boundaries change, the integral may need to be split.

Setting Up the Integral

A reliable setup begins by fixing a height \(y\) and asking what horizontal segment of the region lies there. The left endpoint gives \(x_{\text{left}}(y)\), and the right endpoint gives \(x_{\text{right}}(y)\). Their difference is the shell height. The y-limits come from the bottom and top of the region, not from its leftmost and rightmost x-values.

Worked Examples

A Common Mistake: Omitting the Radius

In the shell formula, the shell height is only one factor. The circumference also depends on the radius, so the integrand must include \(y\) as well as the horizontal width.

For example, take the region \(0\le y\le1\), \(y^2\le x\le2y\). Its shell height is \(2y-y^2\). A common incorrect setup is

This calculation has omitted the radius \(y\). The correct setup includes radius times height:

Differentiating the antiderivative gives \(2y^2-y^3=y(2y-y^2)\), confirming that the corrected integral includes both required factors. The circumference \(2\pi y\) is what makes shells farther from the axis larger; leaving out \(y\) does not account for that change.

Checks Before Evaluating

Before calculating, check that the variable and geometry agree: horizontal slices mean \(dy\); the bounds are y-values; the radius is the distance to the x-axis; and the shell height is a right x-coordinate minus a left x-coordinate. Keeping these roles distinct prevents many setup errors.

Also check that the final integral has the expected units. A radius contributes one unit of length, a shell height contributes another, and \(dy\) contributes a third. The result is therefore in cubic units, as a volume should be.