AP Calculus AB and BC glossary

Axis of revolution

Also called: Axis of rotation

The axis of revolution is the line a plane region is rotated about to sweep out a solid. It fixes every radius in the volume integral, since a radius is the distance from the curve to that axis: rotating about y = 3 gives 3 minus f of x when the curve lies below the line, and f of x minus 3 when it lies above.

A radius is a distance from the axis, never from the $x$ axis by default. For a horizontal axis $y=k$ and a region lying between that line and the curve $y=f(x)$ , each disk has radius $|k-f(x)|$ , which reduces to $f(x)$ only when $k=0$ . Squaring removes the absolute value.

V=\pi\int_a^b\bigl(k-f(x)\bigr)^2\,dx

The axis also decides whether the slice is solid. If it borders the region along the whole interval, the slices are disks. If a gap separates the axis from the near edge of the region, every slice is a washer, and the outer and inner radii must both be measured from that same line. If the axis cuts through the region, the two halves sweep overlapping solids, so neither formula applies as written: measure to the boundary farther from the axis, or split the region at the axis first. A vertical axis such as $x=5$ changes the picture again, since distances are then horizontal.

The mistake

Using $f(x)$ as the radius no matter what the axis is. When the axis is $y=3$ and the curve sits below it, the radius is $3-f(x)$ . Writing $f(x)$ measures from the $x$ axis instead and computes the volume of a different solid entirely.

Appears in: Unit 8: Applications of Integration