Concept preview

What you will be able to do

1. 01Set up a solid-of-revolution volume by slicing a region into thin discs perpendicular to the axis of revolution and identifying the radius $R$ of a typical disc.
2. 02Formulate and evaluate the disc-method integral $V=\pi\int_a^b [R(x)]^2\,dx$ when a region that touches the $x$-axis is revolved about the $x$-axis, and the analogous $y$-integral when revolving about the $y$-axis.
3. 03Explain why the radius must be squared and keep answers exact by leaving $\pi$ symbolic, choosing the correct variable of integration and limits from the geometry.

Begin with a region in the plane bounded above by a curve $y=f(x)$, below by the $x$-axis, and on the sides by the vertical lines $x=a$ and $x=b$. Because the bottom edge of this region lies exactly on the $x$-axis, the region touches the axis of revolution with no gap. Now spin the whole region one full turn about the $x$-axis. Each vertical strip of the region — a thin rectangle of height $f(x)$ and width $dx$ — sweeps out a thin circular disc, like a single coin standing on edge. The disc is centred on the axis, its flat faces are perpendicular to the axis, its radius is the height of the strip, and its thickness is $dx$.