<p>Let&nbsp;\(\mathrm{f}\) and \(\mathrm{g}\) be two functions defined on \([a, b]\) with \(\mathrm{f}(x) \geq \mathrm{g}(x) \geq 0\) for each \(x\) in \([a, b]\).</p>
<p>Let&nbsp;\(R\) be the region bounded by \(y = \mathrm{f}(x)\), \(y = \mathrm{g}(x)\) and the lines \(x = a\) and \(x = b\).</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/1772233623731-uqz7rsf1y2.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Volume of solid obtained by rotating the region \(R\) about the \(x\)-axis through \(360{}^\circ\)</p>
<p>\[= \int_{a}^{b} \pi [\mathrm{f}(x)]^2 \, \mathrm{d}x - \int_{a}^{b} \pi [\mathrm{g}(x)]^2 \, \mathrm{d}x = \pi \int_{a}^{b} \left( [\mathrm{f}(x)]^2 - [\mathrm{g}(x)]^2 \right) \, \mathrm{d}x\]</p>

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 \(360{}^\circ\)

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

Volume between two curves

Volume between two curves