The Radius Starts at the Axis

You'll learnto measure a radius from whatever line the region is being spun about, to get the order of subtraction from which side the region is on, to keep the cross term the shift creates, and to test the radius where it has to vanish.

Writing r = f(x) works in Topic 8.9 for exactly one reason: the axis was y = 0. Move the axis to y = 2 and the radius becomes the distance from that line to the curve. A radius is a distance from the axis of revolution, which means it is never negative and never starts anywhere else. One ten-second check catches every error here — evaluate the radius where the region touches the axis, because it has to come out zero there.

8 STEPS · 6 QUICK CHECKS · TEST THE RADIUS WHERE IT MUST VANISH · v1

r = |f(x) − k| ⇒ the order comes from which side ⇒ r must vanish where they touch

Before you start

What you're looking at

A region, and a dashed line it is being spun about — this time not one of the coordinate axes. Each radius is drawn as a segment starting on that line and ending on the boundary.

The question

How far is the boundary from that line at this position, and which way round does the subtraction go?

Watch for

Where the region touches the axis, the radius has to be zero. Any expression that does not vanish there is measuring from the wrong line, however plausible it looks.

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