Rotation about x-axis

Consider the region below that is rotated \(360^\circ\) about the \(x\)-axis.

If the shaded area under the curve \(y = \mathrm{f}(x)\) between \(x = a\) and \(x = b\) is rotated through \(360{}^\circ\) or \(2\pi\) about the \(x\)-axis, a solid is generated as shown in Fig \(4\) above.

Volume generated about \(x\)-axis:

\[V = \int_{a}^{b} \pi y^2 \, \mathrm{d}x \quad \text{or} \quad \int_{a}^{b} \pi [\mathrm{f}(x)]^2 \, \mathrm{d}x\]