<p style="font-size: 1.125rem"><strong>VOLUME OF REVOLUTION</strong></p>
<p>REVOLUTION 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/1768613558541-azmxrzxupt.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p style="text-align: center">\(V = \pi \displaystyle\int\nolimits_{a}^{b} [\mathrm{f}(x)]^2 \, \mathrm{d}x = \pi \displaystyle\int\nolimits_{a}^{b} y^2 \, \mathrm{d}x\)</p>
<p>REVOLUTION ABOUT THE \(y\)-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/1768613698066-r2b6vodh1ws.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p style="text-align: center">\(V = \pi \displaystyle\int\nolimits_{a}^{b} [\mathrm{f}(y)]^2 \, \mathrm{d}y = \pi \displaystyle\int\nolimits_{a}^{b} x^2 \, \mathrm{d}y\)</p>
<p style="font-size: 1.125rem"><strong>VOLUME BOUNDED BY TWO CURVES</strong></p>
<p>REVOLUTION 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/1768613947998-g6cqqc0gasu.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p style="text-align: center">\(V = \pi \displaystyle\int\nolimits_{a}^{b} [\mathrm{g}(x)]^2 - [\mathrm{f}(x)]^2 \, \mathrm{d}x = \pi \displaystyle\int\nolimits_{a}^{b} (y_2)^2 - (y_1)^2 \, \mathrm{d}x\)</p>
<p>REVOLUTION ABOUT THE \(y\)-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/1768614161068-p75n3vej87n.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p style="text-align: center">\(V = \pi r^2 (c-a) - \displaystyle\int\nolimits_{a}^{b} [\mathrm{f}(y)]^2 \, \mathrm{d}y - \displaystyle\int\nolimits_{b}^{c} [\mathrm{g}(y)]^2 \, \mathrm{d}y\)</p>
<p style="text-align: center">\(= \pi r^2 (c-a) - \displaystyle\int\nolimits_{a}^{b} (x_1)^2 \, \mathrm{d}y - \displaystyle\int\nolimits_{b}^{c} (x_2)^2 \, \mathrm{d}y\)</p>

\(V = \pi \displaystyle \displaystyle\int\nolimits_{a}^{b} [\mathrm{f}(x)]^2 \, \mathrm{d}x = \pi \displaystyle \displaystyle\int\nolimits_{a}^{b} y^2 \, \mathrm{d}x\)

\(V = \pi \displaystyle \displaystyle\int\nolimits_{a}^{b} [\mathrm{f}(y)]^2 \, \mathrm{d}y = \pi \displaystyle \displaystyle\int\nolimits_{a}^{b} x^2 \, \mathrm{d}y\)

\(V = \pi \displaystyle \displaystyle\int\nolimits_{a}^{b} [\mathrm{g}(x)]^2 - [\mathrm{f}(x)]^2 \, \mathrm{d}x = \pi \displaystyle \displaystyle\int\nolimits_{a}^{b} (y_2)^2 - (y_1)^2 \, \mathrm{d}x\)

\(V = \pi r^2 (c-a) - \displaystyle \displaystyle\int\nolimits_{a}^{b} [\mathrm{f}(y)]^2 \, \mathrm{d}y - \displaystyle \displaystyle\int\nolimits_{b}^{c} [\mathrm{g}(y)]^2 \, \mathrm{d}y\)

\(= \pi r^2 (c-a) - \displaystyle \displaystyle\int\nolimits_{a}^{b} (x_1)^2 \, \mathrm{d}y - \displaystyle \displaystyle\int\nolimits_{b}^{c} (x_2)^2 \, \mathrm{d}y\)

Summary Notes: Volume of Revolution

Summary Notes: Volume of Revolution

Video Solution: