Volume with Disc Method: Revolving Around Other Axes

3. Visual Connection

5. Concept Development

AP focus. The disk method still uses \(V=\int A\), but an axis such as \(y=k\) or \(x=h\) changes the radius. Radius is the perpendicular distance from the axis of revolution to the boundary of the region, not the boundary's raw coordinate.

Horizontal shifted axis: \(y=k\)

Use vertical slices, which are perpendicular to the horizontal axis, and integrate with respect to \(x\). If the region runs from \(y=f(x)\) all the way to \(y=k\), each slice becomes a disk:

If the graph is above the axis, write \(R=f(x)-k\); if it is below, write \(R=k-f(x)\). The square gives the same area, but writing a positive distance makes the geometry easier to check.

Vertical shifted axis: \(x=h\)

Use horizontal slices and integrate with respect to \(y\). Write the boundary as \(x=g(y)\). If the region fills the gap from the curve to the axis, then

A boundary to the right of \(x=h\) gives \(R=g(y)-h\); a boundary to the left gives \(R=h-g(y)\).

When is the cross section a disk?

Draw one segment perpendicular to the axis. Its rotation creates a disk only when the segment reaches the axis, so the inner radius is zero. If a gap remains, the cross section has a hole and requires the washer method. If a slice crosses the axis and extends to both sides, use the farther endpoint as the disk radius; split the integral if the farther endpoint changes.

Setup workflow

1. Sketch and label the shifted axis.
2. Choose slices perpendicular to that axis: \(dx\) for \(y=k\), \(dy\) for \(x=h\).
3. Confirm that every slice reaches the axis.
4. Write radius as a distance, then form \(A=\pi R^2\).
5. Use bounds in the same variable as the differential.
6. Evaluate and report cubic units.

Numerical data. When radii are given in a table, first convert each radius to disk area \(A=\pi R^2\), then apply a numerical integration rule to the areas. Squaring an average radius is generally incorrect.

6. Detailed Worked Example and Error Check

Example 1: A horizontal axis above the region

Revolve the region between \(y=x^2\) and \(y=4\), \(-2\le x\le2\), around \(y=4\). Vertical slices reach the axis, and the distance from \(x^2\) to \(4\) is \(R(x)=4-x^2\):

Using \(x^2\) as the radius would measure from the \(x\)-axis, not from the stated axis.

Example 2: A horizontal axis below the region

The region between \(y=x^2\) and \(y=-1\) for \(-1\le x\le1\) is revolved around \(y=-1\). The vertical distance is \(R(x)=x^2-(-1)=x^2+1\):

The parentheses preserve the complete distance before it is squared.

Example 3: A boundary below a shifted horizontal axis

Revolve the region between \(y=-x\) and \(y=1\), \(0\le x\le2\), around \(y=1\). Since the graph lies below the axis, \(R(x)=1-(-x)=1+x\):

Example 4: A shifted vertical axis

Revolve the region between \(x=y^2\) and \(x=4\), \(-2\le y\le2\), around \(x=4\). Horizontal slices create disks with radius \(R(y)=4-y^2\):

The bounds are \(y\)-values because the slices have thickness \(dy\).

Example 5: A vertical axis left of the boundary

The region between \(x=y\) and \(x=-1\) for \(-1\le y\le2\) is revolved around \(x=-1\). The radius is \(R(y)=y-(-1)=y+1\):

The result also agrees with the cone formula using height \(3\) and radius \(3\).

Example 6: A general cone about \(y=k\)

For \(m>0\), revolve the region between \(y=k+mx\) and \(y=k\) on \([0,L]\) around \(y=k\). The shift \(k\) cancels when distance is computed:

This is \(\tfrac13\pi r^2h\) with \(r=mL\) and \(h=L\).

Example 7: Decide between disks and washers

The region between \(y=x^2\) and \(y=x^2+1\) is revolved around \(y=-2\). A vertical slice begins at distance \(x^2+2\), so it never reaches the axis. The cross section has a hole and is a washer, not a disk. This topic's formula \(\pi R^2\) alone cannot represent that cross section.

Example 8: Approximate from tabular radii

For disks about a shifted axis, suppose \(R(0)=2\), \(R(1)=3\), and \(R(3)=1\). The corresponding areas are \(4\pi,9\pi,\pi\). Applying the trapezoidal rule to area gives

Common errors

• Using \(f(x)\) instead of the distance \(|f(x)-k|\).
• Writing \(\pi f(x)^2-k^2\) instead of \(\pi[f(x)-k]^2\).
• Choosing slices parallel rather than perpendicular to the axis.
• Using \(dx\) for disks around a vertical axis without rewriting the region.
• Using disks even though the region leaves a gap around the axis.
• Applying a numerical rule directly to radii instead of to \(\pi R^2\).
• Forgetting cubic units.