Locate the shifted rotation axis

A region turns about the line y = 3

The teal region lies between the curve \(y = \sqrt{x}\) and the line \(y = 3\), for \(0 \le x \le 9\). The dashed line is the axis of rotation, the line that the region turns about. Every point on it has height 3, so its equation is \(y = 3\).

Press "Turn the region". The region sweeps out a solid around the dashed line, not around the x-axis.

Move the slider. The teal strip is a slice of the region, cut at one value of x and a small step in x thick. It meets the dashed line at a right angle and sweeps the teal disk centered on that line.

The axis is a horizontal or a vertical line

The region can have any shape, if one edge lies along the axis. The dashed line is horizontal, so its equation is \(y = k\), where \(k\) is the height where it crosses the y-axis. A vertical axis has an equation \(x = h\), where \(h\) is the position where it crosses the x-axis.

Move the slice. It meets the dashed line at a right angle, and the solid forms around that line, wherever the line is.

Notation

1. A slice meets the axis at a right angle. About a horizontal axis \(y = k\) the slices are vertical, so the variable is \(x\).

   \[\cy{y = k}\ \Rightarrow\ dx\]

2. About a vertical axis \(x = h\) the slices are horizontal for the same reason, so the variable is \(y\).

   \[\cy{x = h}\ \Rightarrow\ dy\]

   Example 1 shows a vertical axis.

3. The x-axis is the line \(y = 0\), and the y-axis is the line \(x = 0\). Earlier work about a coordinate axis is the case \(k = 0\) or \(h = 0\).

4. The orange segment is the radius of the disk, measured from the axis to the curve, so you find the axis before you measure anything.

When it fails

A disk needs the region to touch the axis. Press the button to move the region off the line.

A gap opens between the axis and the region, so each slice sweeps a ring with a hole in its center. The second button shows the hole red: a disk formula would count it as solid, so this lesson covers regions that touch the axis.

Example 1: a vertical axis

The region between the parabola \(x = y^2\) and the line \(x = 4\) turns about the line \(x = 4\). Find the axis, the direction of the slices, the variable and its bounds.

1. Every point on the line \(x = 4\) has x equal to 4, so the line is vertical. It is the right edge of the region.

   \[\cy{x = 4}\]

2. A slice meets the axis at a right angle, so the slices are horizontal. Each slice is a small step \(dy\) thick, so the variable is \(y\).

   \[dy\]

3. The parabola meets the axis where \(y^2 = 4\), so y runs from \(-2\) to \(2\).

   \[-2 \le y \le 2\]

Example 2: not the y-axis

The region under \(y = x^2\), for \(0 \le x \le 3\), turns about the line \(x = 3\). Find the axis, the slices and the bounds.

1. The line \(x = 3\) is vertical, and it touches the region along its whole right edge.

   \[\cy{x = 3}\]

2. A tempting wrong move: A vertical line is the y-axis, so the region turns about the y-axis.

   \[x = 3\ \text{means the y-axis}\]

   The red dashed line is the y-axis, \(x = 0\), three units to the left. It touches the region only at the origin, but the line \(x = 3\) touches the whole right edge.

3. The axis is the line \(x = 3\). A slice meets it at a right angle, so the slices are horizontal and the variable is \(y\).

4. The curve reaches the axis at \(y = 3^2 = 9\), and the region begins on the x-axis, so y runs from 0 to 9.

   \[0 \le y \le 9\]

Practice

The region is bounded by the y-axis, the line \(y = 6\) and the line \(y = 3x\). It turns about the line \(y = 6\). State the axis, the slices, the variable and the bounds.

Move the slider along the dashed line. The slice meets the line at a right angle.

Questions

A triangle has corners at \((1, 1)\), \((5, 1)\) and \((5, 4)\). It turns about the dashed line along its right edge.

1. What is the equation of the dashed line?

   • \(x = 5\)
   • \(y = 5\)
   • \(x = 0\)
   • \(y = 1\)
   Answer

   \(x = 5\)

2. Which variable and which bounds does the integral for the volume use?

   • \(y\), from 1 to 4
   • \(x\), from 1 to 5
   • \(y\), from 0 to 4
   Answer

   \(y\), from 1 to 4

3. A region lies between the curve \(y = x^2\) and the line \(y = 9\). It turns about the line \(y = 9\). Over which values of x do the slices run?

   • \(-3\) to 3
   • 0 to 9
   • \(-9\) to 9
   Answer

   \(-3\) to 3

Summary

You read the equation of the axis, \(y = k\) or \(x = h\), and place the slices at right angles to that line.

Each slice at a right angle to the axis sweeps a disk centered on that line, wherever the line is.

When a problem names a line other than a coordinate axis, find that line first, before you write a radius.

This is one of four skills in topic 8.10 of Which & Why: AP Calculus.