Volume between two curves

Let \(\mathrm{f}\) and \(\mathrm{g}\) be two functions defined on \([a, b]\) with \(\mathrm{f}(x) \geq \mathrm{g}(x) \geq 0\) for each \(x\) in \([a, b]\).

Let \(R\) be the region bounded by \(y = \mathrm{f}(x)\), \(y = \mathrm{g}(x)\) and the lines \(x = a\) and \(x = b\).

Volume of solid obtained by rotating the region \(R\) about the \(x\)-axis through \(360{}^\circ\)

\[= \int_{a}^{b} \pi [\mathrm{f}(x)]^2 \, \mathrm{d}x - \int_{a}^{b} \pi [\mathrm{g}(x)]^2 \, \mathrm{d}x = \pi \int_{a}^{b} \left( [\mathrm{f}(x)]^2 - [\mathrm{g}(x)]^2 \right) \, \mathrm{d}x\]