Washers Around Shifted Axes

Why a Shifted Axis Changes the Radii

When a region is rotated around a coordinate axis, its washer radii are measured from that axis. If the axis is shifted, the same rule applies, but the distances change: the radii are measured from the shifted line, not from the \(x\)-axis or \(y\)-axis.

For example, a point at height \(y\) is a distance \(|y-c|\) from the horizontal line \(y=c\). A point with horizontal coordinate \(x\) is a distance \(|x-c|\) from the vertical line \(x=c\). These distances determine the radii of the circular cross-sections.

Washer Formula for a Shifted Axis

Assume the region lies entirely on one side of the axis and that each slice perpendicular to the axis runs from a nearer boundary to a farther boundary. Rotating such a slice produces a washer. Its outer radius \(R\) is the distance to the farther boundary, and its inner radius \(r\) is the distance to the nearer boundary. Thus \(0\leq r\leq R\).

The area of a washer is the area of its outer disk minus the area of its inner disk. A disk of radius \(s\) has area \(\pi s^2\), so the washer area is \(\pi(R^2-r^2)\). Adding these cross-sectional areas by integration gives the volume.

For a horizontal axis \(y=c\), a vertical slice at \(x\) has boundary heights \(y_{\mathrm{low}}\) and \(y_{\mathrm{high}}\). Its distances from the axis are \(|y_{\mathrm{low}}-c|\) and \(|y_{\mathrm{high}}-c|\). For a vertical axis \(x=c\), a horizontal slice at \(y\) has boundary coordinates \(x_{\mathrm{left}}\) and \(x_{\mathrm{right}}\), with distances \(|x_{\mathrm{left}}-c|\) and \(|x_{\mathrm{right}}-c|\). In either case, identify which distance is larger before assigning \(R\) and \(r\).

Why the Washer Integral Works

Consider a thin slice of the region whose thickness is \(\Delta x\) when the axis is horizontal. Its rotation is approximately a washer of area \(\pi(R^2-r^2)\) and thickness \(\Delta x\). Its approximate volume is therefore \(\pi(R^2-r^2)\Delta x\). Adding these contributions over the interval gives a Riemann sum. In the limit as the slice widths tend to zero, that sum becomes the definite integral of the cross-sectional area. With a vertical axis, the same reasoning uses thickness \(\Delta y\).

according to the slice direction. The formula is the same as for washers around the coordinate axes; only the distances used for \(R\) and \(r\) have changed.

Worked Examples: Horizontal Shifted Axis

The distances are measured from \(y=3\), even though the boundary curves are described by their \(y\)-coordinates. In particular, the height \(y=x\) does not itself give the inner radius; its distance from the axis is \(3-x\).

Worked Example: Vertical Shifted Axis

A Common Mistake: Using Coordinates as Radii

A boundary coordinate is not automatically a radius. The radius is the distance from the boundary to the axis. In the first example, the axis is \(y=3\), while the slice endpoints are \(y=0\) and \(y=x\). A mistaken setup might treat \(x\) as the inner radius and \(0\) as the outer radius:

This uses distances from the \(x\)-axis, not from \(y=3\). For the actual rotation, the distances are \(3\) and \(3-x\), so the correct setup is

The correction is to locate the axis first, then measure both radii from it. The larger distance is the outer radius, regardless of which boundary supplies it.

A Reliable Setup

This method matters because shifting the axis can change both radii, even when the region itself is unchanged. A correct washer integral depends on the geometry of the slice relative to the actual axis, not merely on the equations describing the region.