<p>Consider the region below that is rotated \(360^\circ\) about the \(x\)-axis.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1772232962100-vnndaa4n9ti.png" style="max-width: min(400px, 100%); height: auto; display: block;" alt="Figure 4" class="aligncenter custom-width" /></figure></div>
<p>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.</p>
<p>Volume generated about \(x\)-axis:</p>
<p style="text-align: center">\[V = \int_{a}^{b} \pi y^2 \, \mathrm{d}x \quad \text{or} \quad \int_{a}^{b} \pi [\mathrm{f}(x)]^2 \, \mathrm{d}x\]</p>

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.

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

Rotation about x-axis

Rotation about x-axis