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<h1 class="center">Solids of Revolution by Shells</h1>
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<p class="center"><img src="images/tree-rings.jpg" alt="Tree Rings are like Shells" height="170" width="500"></p>
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<p>We can have a function, like this one:</p>
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<p class="center"><img src="images/solid-shell-1.svg" alt="Solids of Revolution y=f(x)" height="145" width="196"></p>
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<p>And revolve it around the y-axis to get a solid like this:</p>
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<p class="center"><img src="images/solid-shell-2.svg" alt="Solids of Revolution y=f(x)" height="155" width="202"></p>
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<p>Now, to find its <b>volume</b> we can <b>add up "shells"</b>:</p>
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<p class="center"><img src="images/solid-shell-3.svg" alt="Solids of Revolution y=f(x)" height="164" width="201"></p>
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<p>Each shell has the curved surface area of a <a href="../geometry/cylinder.html">cylinder</a> whose area is <b>2<span class="times">π</span>r</b> times its height:</p>
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<p class="center large"><img src="images/solid-shell-4.svg" alt="Solids of Revolution y=f(x)" height="164" width="201"><br>
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A = 2<span class="times">π</span>(radius)(height)</p>
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<p>And the <b>volume</b> is found by summing all those shells using <a href="integration-introduction.html">Integration</a>:</p>
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<div class="center larger">Volume =
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<div class="intgl">
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<div class="to">b</div>
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<div class="symb"></div>
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<div class="from">a</div>
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</div>2<span class="times">π</span>(radius)(height) dx</div>
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<!-- Volume = INT{a, b} 2 PI (radius)(height) dx -->
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<p class="center">That is our formula for <b>Solids of Revolution by Shells</b></p>
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<p>These are the steps:</p>
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<ul>
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<li>sketch the volume and how a typical shell fits inside it</li>
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<li>integrate <b>2<span class="times">π</span></b> times the <b>shell's radius</b> times the <b>shell's height</b>,</li>
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<li>put in the values for b and a, subtract, and you are done.</li>
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</ul>
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<p>As in this example:</p>
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<div class="example">
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<h3>Example: A Cone!</h3>
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<p>Take the simple function <b>y = b − x</b> between x=0 and x=b</p>
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<p class="center"><img src="images/solid-shell-cone-1.svg" alt="Solids of Revolution y=f(x)" height="123" width="185"></p>
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<p>Rotate it around the y-axis ... and we have a cone!</p>
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<p class="center"><img src="images/solid-shell-cone-2.svg" alt="Solids of Revolution y=f(x)" height="123" width="185"></p>
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<p>Now let us imagine a shell inside:</p>
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<p class="center"><img src="images/solid-shell-cone-3.svg" alt="Solids of Revolution y=f(x)" height="123" width="185"></p>
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<p>What is the shell's radius? It is simply <b>x</b><br>
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What is the shell's height? It is <b>b−x</b></p>
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<p>What is the volume? <b>Integrate 2<span class="times">π</span> times x times (b−x)</b> :</p>
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<div class="center large">Volume =
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<div class="intgl">
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<div class="to">b</div>
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<div class="symb"></div>
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<div class="from">0</div>
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</div>2<span class="times">π</span> x(b−x) dx</div>
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<!-- Volume = INT{0, b} 2 PI x(b−x) dx -->
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/pie-outside.jpg" alt="pie outside" height="93" width="150"></p>
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<p>Now, let's have our <b>pi outside</b> (yum).</p>
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<p>Seriously, we can bring a constant like 2<span class="times">π</span> outside the integral:</p>
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<div style="clear:both"></div>
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<div class="center large">Volume = 2<span class="times">π</span>
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<div class="intgl">
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<div class="to">b</div>
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<div class="symb"></div>
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<div class="from">0</div>
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</div>x(b−x) dx</div>
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<!-- Volume = 2PI INT{0, b} x(b−x) dx -->
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<p>Expand x(b−x) to bx − x<sup>2</sup>:</p>
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<div class="center large">Volume = 2<span class="times">π</span>
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<div class="intgl">
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<div class="to">b</div>
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<div class="symb"></div>
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<div class="from">0</div>
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</div>(bx−x<sup>2</sup>) dx</div>
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<!-- Volume = 2PI INT{0, b} (bx−x^2 ) dx -->
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<p>Using <a href="integration-rules.html">Integration Rules</a> we find the integral of bx − x<sup>2</sup> is:</p>
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<p class="center large"><span class="intbl"><em>bx<sup>2</sup></em><strong>2</strong></span> − <span class="intbl"><em>x<sup>3</sup></em><strong>3</strong></span> + C</p>
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<p>To calculate the <a href="integration-definite.html">definite integral</a> between 0 and b, we calculate the value of the function for <b>b</b> and for <b>0</b> and subtract, like this:</p>
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<div class="tbl">
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<div class="row"><span class="lt">Volume =</span><span class="rtlt">2<span class="times">π</span>(<span class="intbl"><em>b(b)<sup>2</sup></em><strong>2</strong></span> − <span class="intbl"><em>b<sup>3</sup></em><strong>3</strong></span>) − 2<span class="times">π</span>(<span class="intbl"><em>b(0)<sup>2</sup></em><strong>2</strong></span> − <span class="intbl"><em>0<sup>3</sup></em><strong>3</strong></span>)</span></div>
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<div class="row"><span class="lt">=</span><span class="rtlt">2<span class="times">π</span>(<span class="intbl"><em>b<sup>3</sup></em><strong>2</strong></span> − <span class="intbl"><em>b<sup>3</sup></em><strong>3</strong></span>)</span></div>
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<div class="row"><span class="lt">=</span><span class="rtlt">2<span class="times">π</span>(<span class="intbl"><em>b<sup>3</sup></em><strong>6</strong></span>) because <span class="intbl"><em>1</em><strong>2</strong></span> − <span class="intbl"><em>1</em><strong>3</strong></span> = <span class="intbl"><em>1</em><strong>6</strong></span></span></div>
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<div class="row"><span class="lt">=</span><span class="rtlt"><span class="times">π</span><span class="intbl"><em>b<sup>3</sup></em><strong>3</strong></span></span></div>
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</div>
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</div>
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<div class="fun">Compare that result with the more general volume of a <a href="../geometry/cone.html">cone</a>:
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<p class="center"><span class="large">Volume = <span class="intbl">
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<em>1</em>
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<strong>3</strong>
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</span> <span class="times"> π</span> r<sup>2</sup> h</span></p>
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<p>When both <b>r=b</b> and <b>h=b</b> we get:</p>
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<p class="center"><span class="large">Volume = <span class="intbl">
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<em>1</em>
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<strong>3</strong>
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</span> <span class="times"> π</span> b<sup>3</sup></span></p>
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<p>As an interesting exercise, why not try to work out the more general case of any value of r and h yourself?</p></div>
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<p> </p>
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<p>We can also rotate about other values, such as x = 4</p>
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<div class="example">
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<h3>Example: y=x, but rotated around x = 4, and only from x=0 to x=3</h3>
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<p>So we have this:</p>
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<p class="center"><img src="images/solid-shell-conem1-1.svg" alt="Solids of Revolution y=f(x)" height="121" width="179"></p>
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<p>Rotated about x = 4 it looks like this:</p>
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<p class="center"><img src="images/solid-shell-conem1-2.svg" alt="Solids of Revolution y=f(x)" height="125" width="179"><br>
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It is a cone, but with a hole down the center</p>
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<p>Let's draw in a sample shell so we can work out what to do:</p>
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<p class="center"><img src="images/solid-shell-conem1-3.svg" alt="Solids of Revolution y=f(x)" height="116" width="189"></p>
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<p>What is the shell's radius? It is <b>4−x</b> <i>(not just x, as we are rotating around x=4)</i><br>
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What is the shell's height? It is <b>x</b></p>
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<p>What is the volume? <b>Integrate 2<span class="times">π</span> times (4−x) times x</b> :</p>
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<div class="center large">Volume =
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<div class="intgl">
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<div class="to">3</div>
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<div class="symb"></div>
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<div class="from">0</div>
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</div>2<span class="times">π</span>(4−x)x dx</div>
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<!-- Volume = INT{0, 3} 2PI(4−x)x dx -->
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<p><b>2<span class="times">π</span> outside</b>, and expand <b>(4−x)x</b> to <b>4x − x<sup>2</sup></b> :</p>
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<div class="center large">Volume = 2<span class="times">π</span>
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<div class="intgl">
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<div class="to">3</div>
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<div class="symb"></div>
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<div class="from">0</div>
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</div>(4x−x<sup>2</sup>) dx</div>
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<!-- Volume = 2PI INT{0, 3} (4x−x^2) dx -->
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<p>Using <a href="integration-rules.html">Integration Rules</a> we find the integral of 4x − x<sup>2</sup> is:</p>
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<p class="center large"><span class="intbl"><em>4x<sup>2</sup></em><strong>2</strong></span> − <span class="intbl"><em>x<sup>3</sup></em><strong>3</strong></span> + C</p>
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<p>And going between <b>0</b> and <b>3</b> we get:</p>
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<p class="center large">Volume = 2<span class="times">π</span>(<span class="intbl"><em>4(3)<sup>2</sup></em><strong>2</strong></span> − <span class="intbl"><em>3<sup>3</sup></em><strong>3</strong></span>) − 2<span class="times">π</span>(<span class="intbl"><em>4(0)<sup>2</sup></em><strong>2</strong></span> − <span class="intbl"><em>0<sup>3</sup></em><strong>3</strong></span>)</p>
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<p class="center large">= 2<span class="times">π</span>(18−9)</p>
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<p class="center larger">= 18<span class="times">π</span></p>
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</div>
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<p>We can have more complex situations:</p>
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<div class="example">
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<h3>Example: From y=x down to y=x<sup>2</sup></h3>
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<p class="center"><img src="images/solid-shell-d-1.svg" alt="Solids of Revolution about Y" height="183" width="237"></p>
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<p>Rotate around the y-axis:</p>
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<p class="center"><img src="images/solid-shell-d-2.svg" alt="Solids of Revolution about Y" height="183" width="237"></p>
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<p>Let's draw in a sample shell:</p>
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<p class="center"><img src="images/solid-shell-d-3.svg" alt="Solids of Revolution about Y" height="183" width="237"></p>
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<p>What is the shell's radius? It is simply <b>x</b><br>
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What is the shell's height? It is <b>x − x<sup>2</sup></b></p>
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<p>Now <b>integrate 2<span class="times">π</span> times x times x − x<sup>2</sup></b>:</p>
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<div class="center large">Volume =
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<div class="intgl">
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<div class="to">b</div>
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<div class="symb"></div>
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<div class="from">a</div>
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</div>2<span class="times">π</span> x(x − x<sup>2</sup>) dx</div>
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<!-- Volume = INT{a, b} 2PI x(x − x^2) dx -->
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<p>Put 2<span class="times">π</span> outside, and expand x(x−x<sup>2</sup>) into x<sup>2</sup>−x<sup>3</sup> :</p>
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<div class="center large">Volume = 2<span class="times">π</span>
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<div class="intgl">
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<div class="to">b</div>
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<div class="symb"></div>
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<div class="from">a</div>
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</div>(x<sup>2</sup> − x<sup>3</sup>) dx</div>
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<!-- Volume = 2PI INT{a, b} (x^2 − x^3 ) dx -->
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<p>The integral of x<sup>2</sup> − x<sup>3</sup> is <b><span class="intbl"><em>x<sup>3</sup></em><strong>3</strong></span> − <span class="intbl"><em>x<sup>4</sup></em><strong>4</strong></span></b></p>
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<p>Now calculate the volume between a and b ... but what<i> is</i> a and b? a is 0, and b is where x crosses x<sup>2</sup>, which is 1</p>
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<div class="tbl">
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<div class="row"><span class="lt">Volume =</span><span class="rtlt">2<span class="times">π</span> ( <span class="intbl"><em>1<sup>3</sup></em><strong>3</strong></span> − <span class="intbl"><em>1<sup>4</sup></em><strong>4</strong></span> ) − 2<span class="times">π</span> ( <span class="intbl"><em>0<sup>3</sup></em><strong>3</strong></span> − <span class="intbl"><em>0<sup>4</sup></em><strong>4</strong></span> )</span></div>
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||
<div class="row"><span class="lt">=</span><span class="rtlt">2<span class="times">π</span> (<span class="intbl"><em>1</em><strong>12</strong></span>)</span></div>
|
||
<div class="row"><span class="lt">=</span><span class="rtlt"><span class="intbl"><em><span class="times">π</span></em><strong>6</strong></span></span></div>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
<h2>In summary:</h2>
|
||
|
||
<ul>
|
||
<li>Draw the shell so you know what is going on</li>
|
||
<li><b>2<span class="times">π</span></b> outside the integral</li>
|
||
<li>Integrate the <b>shell's radius</b> times the <b>shell's height</b>,</li>
|
||
<li>Subtract the lower end from the higher end</li>
|
||
</ul>
|
||
<p> </p>
|
||
|
||
<div class="related">
|
||
<a href="solids-revolution-disk-washer.html">Solids of Revolution by Disks and Washers</a>
|
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<a href="index.html">Calculus Index</a>
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