Rotate a region adjoining the horizontal axis through a full revolution. A thin vertical strip sweeps out a disk whose radius is its distance from that axis.
\(V=\pi\int_a^b[\mathrm{f}(x)]^2\,\mathrm{d}x\). The cross-sectional area is \(\pi r^2\).
The region under \(y=\sqrt{x}\) from \(x=0\) to \(x=4\) gives:
\(V=\pi\int_0^4(\sqrt{x})^2\,\mathrm{d}x=\pi\int_0^4x\,\mathrm{d}x=8\pi\).
Square the radius before integrating. This is not \(\pi\left(\int_a^b\mathrm{f}(x)\,\mathrm{d}x\right)^2\). Give volume in cubic units.
If the generating region crosses the axis, inspect each cross-section: overlapping rotations must be counted only once.
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