Choose slices perpendicular to the axis
A region turns into a solid
The teal region lies under the curve \(y = x^2\) from \(x = 0\) to \(x = 2\). Press "Turn the region" to turn it once about the dashed x-axis.
The dashed line is the axis of rotation, because the region turns about it. The region sweeps out the solid that the faint outline shows, called a solid of revolution.
A thin vertical strip of the region, called a slice, stands at \(x\). When the region turns, the strip sweeps a flat circle, a disk, at a right angle to the x-axis.
Drag the orange dot along the axis, or move the slider. Each position gives one disk, and the solid is the stack of all these disks.
The slices follow the axis
The region touches the dashed axis of rotation. The teal strip stands across the axis at a right angle, and when the region turns, it sweeps a thin disk. The thickness of the strip and of the disk, \(\cb{dx}\), is a small step along the axis.
Press "Turn about the y-axis". The strip is now horizontal, because only a horizontal strip meets the y-axis at a right angle. Its thickness is \(\cb{dy}\), a small step along the y-axis.
In both cases the slice is perpendicular to the axis of rotation, so its thickness is a step along that axis.
Notation
If the axis is the x-axis, draw vertical slices, because only a vertical strip meets the x-axis at a right angle. Each slice has thickness \(\cb{dx}\).
The thickness runs along the axis, so the integration variable, which names each slice, is the coordinate along the axis, here \(x\).
If the axis is the y-axis, draw horizontal slices. Each slice has thickness \(\cb{dy}\), so the variable is \(y\).
The bounds, the limits of the integral, are the smallest and the largest value of the variable on the region. About the y-axis they are \(\cy{c}\) and \(\cy{d}\), and about the x-axis they are \(\cy{a}\) and \(\cy{b}\).
\[\cy{a} \le x \le \cy{b} \qquad \cy{c} \le y \le \cy{d}\]
When it fails
A disk is solid from the axis to its edge, so the region must reach the axis. Press the button to lift the region off the x-axis.
The region no longer reaches the axis, so each slice sweeps a ring, with an empty center that the second button fills red. A ring is not a disk, so this lesson keeps to regions that reach the axis.
Example 1: turn about the x-axis
The region lies under \(y = \sin x\) from \(x = 0\) to \(x = \pi\). It turns about the x-axis. State the direction of the slices, the variable and the bounds.
The axis is the x-axis, and a slice must meet it at a right angle, so the slices are vertical.
Each slice has thickness \(dx\), so the integration variable is \(x\).
\[\cb{dx}\]The curve meets the x-axis where \(\sin x = 0\), at \(x = 0\) and at \(x = \pi\), so these are the bounds.
\[\cy{0} \le x \le \cy{\pi}\]
Example 2: turn about the y-axis
The region lies in the first quadrant, between the axes and the curve \(y = 9 - x^2\). It turns about the y-axis. State the direction of the slices, the variable and the bounds.
The axis is the y-axis. The curve meets the x-axis at \(x = 3\) and the y-axis at \(y = 9\).
\[9 - 3^2 = 0, \quad 9 - 0^2 = 9\]A tempting wrong move: The strips under the curve are vertical, so integrate in \(x\) from 0 to 3.
\[\int_0^3 \cdots\,dx\]A vertical strip is parallel to the y-axis, so when it turns it sweeps a thin hollow tube, shown in red, not a disk.
Only a horizontal strip meets the y-axis at a right angle, so the slices are horizontal. Each has thickness \(\cb{dy}\), so the variable is \(y\).
\[\cb{dy}\]The region runs from the x-axis up to the top of the curve, so the bounds are 0 and 9.
\[\cy{0} \le y \le \cy{9}\]
Practice
The region is bounded by the y-axis, the line \(y = 3\) and the curve \(x = y^2\). It turns about the y-axis. State the direction of the slices, the variable and the bounds.
Move the slider along the y-axis. The slice that meets the axis at a right angle is horizontal.
Solution
The axis is the y-axis, so the slices are horizontal. The thickness is \(dy\), and the variable is \(y\).
\[\cb{dy}\]The curve starts at the origin and reaches the line \(y = 3\) at \(x = 9\), so the region lies between \(y = 0\) and \(y = 3\).
\[\cy{0} \le y \le \cy{3}\]
Questions
The figure shows a triangle with corners at \((0,0)\), \((4,0)\) and \((0,4)\). Strip A is vertical and strip B is horizontal.
The triangle turns about the x-axis. Which strip becomes a disk?
- Strip A
- Strip B
- Both strips
Answer
Strip A
The triangle now turns about the y-axis. Which variable and which bounds does the integral use?
- \(y\), from 0 to 4
- \(x\), from 0 to 4
- \(y\), from 0 to 1
Answer
\(y\), from 0 to 4
A region is bounded by the axes and the line \(2x + y = 10\). It turns about the y-axis. Over which values does the integration variable run?
- \(y\) from 0 to 10
- \(x\) from 0 to 5
- \(y\) from 0 to 5
Answer
\(y\) from 0 to 10
Summary
You place each slice at a right angle to the axis of rotation, and you integrate in the variable that runs along the axis.
This works because a strip sweeps a disk only when it meets the axis at a right angle, and the integral adds these disks.
Choose the slices first in every volume of revolution, before you write a radius.
This is one of four skills in topic 8.9 of Which & Why: AP Calculus.