When the Axis Moves, the Radii Trade Places

You'll learnto measure both radii from the same shifted axis, to recognise that outer means farther rather than upper, to read a negative volume as a missed swap, and to test both radii where the boundaries meet.

Two radii, one shifted axis, and one fact that catches almost everybody: when the axis sits beyond the region, the boundary that was on top becomes the inner radius. Outer means farther from the axis, and that stops meaning upper the moment the axis moves. The other error here is quieter — adjusting one radius for the shift and leaving the other measured from the coordinate axis, which produces a well-formed integral describing half of one solid and half of another.

8 STEPS · 6 QUICK CHECKS · BOTH RADII, SAME AXIS, SAME FORM · v1

R and r are both |boundary − k| ⇒ outer means FARTHER ⇒ a negative volume is a missed swap

Before you start

What you're looking at

A region between two curves, and a dashed line above them that it is being spun about. Two radius segments start on that line, one reaching each boundary.

The question

Which of the two segments is longer? That boundary supplies the outer radius, whether or not it is the upper curve.

Watch for

Both radii carry the same shift. If one expression contains the axis and the other does not, half the setup is describing a different solid.

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