Similarly, if the shaded area under the curve \(x = \mathrm{f}(y)\) between \(y = a\) and \(y = b\) is rotated through \(360{}^\circ\) or \(2\pi\) about the \(y\)-axis,
Volume of solid generated about \(y\)-axis:
\[V = \int_{a}^{b} \pi x^2 \, \mathrm{d}y \quad \text{or} \quad \int_{a}^{b} \pi [\mathrm{f}(y)]^2 \, \mathrm{d}y\]

Similarly, if the shaded area under the curve \(x = \mathrm{f}(y)\) between \(y = a\) and \(y = b\) is rotated through \(360{}^\circ\) or \(2\pi\) about the \(y\)-axis,

Volume of solid generated about \(y\)-axis:

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

Rotation about y-axis

Rotation about y-axis