Volume with Washer Method: Revolving Around Other Axes

3. Visual Connection

5. Concept Development

AP focus. The washer formula does not change when the axis moves, but both radii do. For an axis \(y=k\) or \(x=h\), measure perpendicular distances from that line:

The axis shift belongs inside every radius before squaring. Squaring removes a distance's sign; it does not erase the shift.

Horizontal axis \(y=k\)

Use vertical slices and \(dx\). Suppose \(f(x)\ge g(x)\).

• If the axis is below both graphs, \(k\le g(x)\), then \(R=f(x)-k\) and \(r=g(x)-k\).
• If the axis is above both graphs, \(k\ge f(x)\), then the lower graph is farther: \(R=k-g(x)\) and \(r=k-f(x)\).

Vertical axis \(x=h\)

Use horizontal slices and \(dy\), writing the boundaries as \(x=F(y)\) and \(x=G(y)\) with \(F(y)\ge G(y)\).

• If the axis is left of both boundaries, \(h\le G(y)\), then \(R=F(y)-h\) and \(r=G(y)-h\).
• If the axis is right of both boundaries, \(h\ge F(y)\), then \(R=h-G(y)\) and \(r=h-F(y)\).

Check the cross-section type

A washer requires a positive gap between the axis and the entire slice. If the shifted axis lies inside a slice, rotation fills the center and the cross section is a disk with \(r=0\). If the axis enters or leaves the region as the variable changes, split the integral where that happens. Also split wherever the farther boundary changes.

Distance-first workflow

1. Draw and label the region and shifted axis.
2. Draw a representative slice perpendicular to the axis.
3. Measure both endpoint distances from the axis.
4. Choose the larger distance as \(R\) and the smaller as \(r\).
5. Write \(A=\pi(R^2-r^2)\) with complete parentheses.
6. Find bounds and all points where the geometry changes.
7. Integrate and report cubic units.

Quick check. Since \(R\ge r\), the integrand \(\pi(R^2-r^2)\) must be nonnegative. A negative expression signals reversed radii or a missing split.

Numerical data. At each listed input, compute the full washer area \(A_i=\pi(R_i^2-r_i^2)\), then approximate \(\int A\). Do not separately average the radii and square them.

6. Detailed Worked Example and Error Check

Example 1: Axis below the region

Revolve the region between \(y=x\) and \(y=x^2\) on \([0,1]\) around \(y=-1\). Since \(x\ge x^2\), the line is farther from the axis:

Example 2: Axis above the same region

Now revolve the same region around \(y=2\). The lower parabola is farther from the axis, so \(R=2-x^2\) and \(r=2-x\):

The same planar region can generate a different volume when the axis moves.

Example 3: Vertical axis left of the region

For \(0\le y\le1\), revolve the region between \(x=y\) and \(x=y^2\) around \(x=-1\). Horizontal slices give

Example 4: Vertical axis right of the region

Revolve that region around \(x=2\). The left boundary \(x=y^2\) is farther from the axis:

Example 5: Constant radii and a geometry check

Revolve the rectangle \(0\le x\le4\), \(1\le y\le3\), around \(y=-2\). The radii are \(R=5\) and \(r=3\):

This agrees with a length-\(4\) outer cylinder of radius \(5\) minus an inner cylinder of radius \(3\).

Example 6: Radius order changes

Rotate the region between \(y=x\) and \(y=1\), \(0\le x\le2\), around \(y=3\). The curves meet at \(x=1\). Before that point \(y=x\) is farther from the axis; afterward \(y=1\) is farther:

Example 7: The shifted axis crosses every slice

Revolve the region between \(y=x^2\) and \(y=2\), \(0\le x\le1\), around \(y=1\). Each vertical slice contains the axis. Rotation fills the center, so these are disks, not washers. The upper endpoint is always distance \(1\), while the lower endpoint is distance \(1-x^2\le1\); therefore

Example 8: Approximate shifted-axis washers from data

At \(x=0,2,4\), suppose \((R,r)=(4,2),(3,1),(2,1)\). The washer areas are \(12\pi,8\pi,3\pi\). The trapezoidal rule gives

Common errors

• Adding or subtracting the shift in only one radius.
• Using the upper graph as \(R\) when the axis is above the region.
• Using the right graph as \(R\) when the axis is to the right of the region.
• Writing \([f(x)]^2-k^2\) instead of \([f(x)-k]^2\).
• Using \(dx\) for washers perpendicular to a vertical axis.
• Failing to split when the radius order or cross-section type changes.
• Using washers when a slice crosses the axis and creates a disk.
• Allowing \(R^2-r^2\) to be negative without correcting the geometry.