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<h1 class="center">Solids of Revolution by Disks</h1>
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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-rev-1.svg" alt="Solids of Revolution y=f(x)" height="89" width="316" ></p>
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<p>And revolve it around the x-axis like this:</p>
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<p class="center"><img src="images/solid-rev-2.svg" alt="Solids of Revolution y=f(x)" height="135" width="316" ></p>
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<p>To find its <b>volume</b> we can <b>add up a series of disks</b>:</p>
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<p class="center"><img src="images/solid-rev-3.svg" alt="Solids of Revolution y=f(x)" height="135" width="316" ></p>
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<p>Each disk's face is a circle:</p>
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<p class="center large"><img src="images/solid-rev-4.svg" alt="Solids of Revolution y=f(x)" height="135" width="316" ></p>
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<p>The <a href="../geometry/circle-area.html">area of a circle</a> is <span class="times">π</span> times radius squared:</p>
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<p class="center large">A = <span class="times">π</span> r<sup>2</sup></p>
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<p>And the radius <b>r</b> is the value of the function at that point <b>f(x)</b>, so:</p>
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<p class="center large">A = <span class="times">π</span> f(x)<sup>2</sup></p>
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<p>And the <b>volume</b> is found by summing all those disks 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> <span class="times">π</span> f(x)<sup>2</sup> dx</div>
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<!-- Volume = INT{a, b} PI f(x)^2 dx -->
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<p class="center">And that is our formula for <b>Solids of Revolution by Disks</b></p>
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<p>In other words, to find the volume of revolution of a function f(x): <b>integrate pi times the square of the function</b>.</p>
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<div class="example">
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<h3>Example: A Cone</h3>
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<p>Take the very simple function <b>y=x</b> between 0 and b</p>
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<p class="center"><img src="images/solid-rev-cone-1.svg" alt="Solids of Revolution y=f(x)" height="90" width="140" ></p>
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<p>Rotate it around the x-axis ... and we have a cone!</p>
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<p class="center"><img src="images/solid-rev-cone-2.svg" alt="Solids of Revolution y=f(x)" height="139" width="141" ></p>
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<p>The radius of any disk is the function f(x), which in our case is simply <b>x</b></p>
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<p class="center"><img src="images/solid-rev-cone-3.svg" alt="Solids of Revolution y=f(x)" height="139" width="141" ></p>
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<p>What is its volume? <b>Integrate pi times the square of the function 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><span class="times">π</span> x<sup>2</sup> dx</div>
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<!-- Volume = INT{0, b} PI x^2 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>First, let's have our <b>pi outside</b> (yum).</p>
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<p>Seriously, it is OK to bring a constant outside the integral:</p>
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<div style="clear:both"></div>
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<div class="center large">Volume = <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<sup>2</sup> dx</div>
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<!-- Volume = PI INT{0, b} x^2 dx -->
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<p>Using <a href="integration-rules.html">Integration Rules</a> we find the integral of x<sup>2</sup> is: <b><span class="intbl"><em>x<sup>3</sup></em><strong>3</strong></span> + C</b></p>
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<p>To calculate this <a href="integration-definite.html">definite integral</a>, we calculate the value of that function for <b>b</b> and for <b>0</b> and subtract, like this:</p>
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<p class="center larger">Volume = <span class="times"><b>π</b></span> (<span class="intbl"><em>b<sup>3</sup></em><strong>3</strong></span> − <span class="intbl"><em>0<sup>3</sup></em><strong>3</strong></span>)</p>
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<p class="center larger">= <span class="times">π</span> <span class="intbl"><em>b<sup>3</sup></em><strong>3</strong></span></p>
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</div>
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<div class="fun"><p>Compare that result with the more general volume of a <a href="../geometry/cone.html">cone</a>:</p>
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<p class="center"><span class="larger">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="larger">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 lines, such as x = −1</p>
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<div class="example">
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<h3>Example: Our Cone, But About x = −1</h3>
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<p>So we have this:</p>
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<p class="center"><img src="images/solid-rev-conem1-1.svg" alt="Solids of Revolution y=f(x)" height="136" width="163" ></p>
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<p>Rotated about x = −1 it looks like this:</p>
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<p class="center"><img src="images/solid-rev-conem1-2.svg" alt="Solids of Revolution y=f(x)" height="169" width="168" ><br>
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The cone is now bigger, with its sharp end cut off (a <i>truncated cone</i>)</p>
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<p>Let's draw in a sample disk so we can work out what to do:</p>
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<p class="center"><img src="images/solid-rev-conem1-3.svg" alt="Solids of Revolution y=f(x)" height="169" width="168" ></p>
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<p>OK. Now what is the radius? It is our function <b>y=x</b> plus an extra <b>1</b>:</p>
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<p class="center large">y = x + 1</p>
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<p>Then <b>integrate pi times the square of that function</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><span class="times">π</span> (x+1)<sup>2</sup> dx</div>
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<!-- Volume = INT{0, b}PI (x+1)^2 dx -->
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<p><b>Pi outside</b>, and expand (x+1)<sup>2</sup> to x<sup>2</sup>+2x+1 :</p>
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<div class="center large">Volume = <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<sup>2</sup> + 2x + 1) dx</div>
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<!-- Volume = PI INT{0, b}(x^2 + 2x + 1) dx -->
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<p>Using <a href="integration-rules.html">Integration Rules</a> we find the integral of x<sup>2</sup>+2x+1 is <b>x<sup>3</sup>/3 + x<sup>2</sup> + x + C</b></p>
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<p>And going between <b>0</b> and <b>b</b> we get:</p>
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<p class="center larger">Volume = <span class="times">π</span> (b<sup>3</sup>/3+b<sup>2</sup>+b − (0<sup>3</sup>/3+0<sup>2</sup>+0))</p>
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<p class="center larger">= <span class="times">π</span> (b<sup>3</sup>/3+b<sup>2</sup>+b)</p>
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</div>
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<p>Now for another type of function:</p>
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<div class="example">
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<h3>Example: The Square Function</h3>
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<p>Take <b>y = x<sup>2</sup></b> between x=0.6 and x=1.6</p>
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<p class="center"><img src="images/solid-rev-x2-1.svg" alt="Solids of Revolution y=x^2" height="155" width="140" ></p>
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<p>Rotate it around the x-axis:</p>
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<p class="center"><img src="images/solid-rev-x2-2.svg" alt="Solids of Revolution y=x^2" height="261" width="146" ></p>
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<p>What is its volume? <b>Integrate pi times the square of 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">1.6</div>
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<div class="symb"></div>
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<div class="from">0.6</div>
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</div><span class="times">π</span> (x<sup>2</sup>)<sup>2</sup> dx</div>
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<!-- Volume = INT{0.6, 1.6}PI (x^2 )^2 dx -->
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<p>Simplify by having pi outside, and also (x<sup>2</sup>)<sup>2</sup> = x<sup>4</sup> :</p>
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<div class="center large">Volume = <span class="times">π</span>
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<div class="intgl">
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<div class="to">1.6</div>
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<div class="symb"></div>
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<div class="from">0.6</div>
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</div>x<sup>4</sup> dx</div>
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<!-- Volume = PI INT{0.6, 1.6}x^4 dx -->
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<p>The integral of x<sup>4</sup> is <b>x<sup>5</sup>/5 + C</b></p>
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<p>And going between 0.6 and 1.6 we get:</p>
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<p class="center">Volume = <span class="times">π</span> ( 1.6<sup>5</sup>/5 − 0.6<sup>5</sup>/5 )</p>
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<p class="center">≈ <font face="Times New Roman, Times, serif" size="+1"></font>6.54</p>
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</div>
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<p>Can you rotate <b>y = x<sup>2</sup></b> about x = −1 ?</p>
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<h2>In summary:</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/pie-outside.jpg" alt="pie outside" height="93" width="150" ></p>
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<ul>
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<li>Have pi outside</li>
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<li>Integrate the <b>function squared</b></li>
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<li>Subtract the lower end from the higher end</li>
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</ul>
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<p> </p>
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<h2>About The Y Axis</h2>
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<p>We can also rotate about the Y axis:</p>
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<div class="example">
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<h3>Example: The Square Function</h3>
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<p>Take y=x<sup>2</sup>, but this time using the <b>y-axis</b> between y=0.4 and y=1.4</p>
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<p class="center"><img src="images/solid-rev-y-1.svg" alt="Solids of Revolution about Y" height="145" width="161" ></p>
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<p>Rotate it around the <b>y-axis</b>:</p>
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<p class="center"><img src="images/solid-rev-y-2.svg" alt="Solids of Revolution about Y" height="146" width="196" ></p>
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<p>And now we want to integrate in the y direction!</p>
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<p>So we want something like <b>x = g(y)</b> instead of y = f(x). In this case it is:</p>
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<p class="center large">x = √(y)</p>
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<p>Now <b>integrate pi times the square of √(y)<sup>2</sup></b> (and dx is now <b>dy</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">1.4</div>
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<div class="symb"></div>
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<div class="from">0.4</div>
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</div><span class="times">π</span> √(y)<sup>2</sup> dy</div>
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<!-- Volume = INT{0.4, 1.4} PI SQR(y)^2 dy -->
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<p>Simplify with pi outside, and √(y)<sup>2</sup> = y :</p>
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<div class="center large">Volume = <span class="times">π</span>
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<div class="intgl">
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<div class="to">1.4</div>
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<div class="symb"></div>
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<div class="from">0.4</div>
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</div>y dy</div>
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<!-- Volume = PI INT{0.4, 1.4} y dy -->
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<p>The integral of y is y<sup>2</sup>/2</p>
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<p>And lastly, going between 0.4 and 1.4 we get:</p>
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<p class="center">Volume = <span class="times">π</span> ( 1.4<sup>2</sup>/2 − 0.4<sup>2</sup>/2 )</p>
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<p class="center">≈ <b>2.83...</b></p>
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</div>
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<p> </p>
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<h2>Washer Method</h2>
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<p class="center"><img src="../geometry/images/washers.jpg" alt="Washers (various)" height="117" width="300" ><br>
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Washers: Disks with Holes</p>
|
||
<p>What if we want the volume <b>between two functions</b>?</p>
|
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<div class="example">
|
||
|
||
<h3>Example: Volume between the functions <b>y=x</b> and <b>y=x<sup>3</sup></b> from x=0 to 1</h3>
|
||
<p>These are the functions:</p>
|
||
<p class="center"><img src="images/solid-rev-x-x3-1.svg" alt="Solids of Revolution between y=x and y=x^3" height="155" width="140" ></p>
|
||
<p>Rotated around the x-axis:</p>
|
||
<p class="center"><img src="images/solid-rev-x-x3-2.svg" alt="Solids of Revolution between y=x and y=x^3" height="219" width="207" ></p>
|
||
<p>The disks are now "washers":</p>
|
||
<p class="center"><img src="images/solid-rev-x-x3-3.svg" alt="Solids of Revolution between y=x and y=x^3" height="219" width="207" ></p>
|
||
<p>And they have the area of an <a href="../geometry/annulus.html">annulus</a>:</p>
|
||
<p class="center"><img src="../geometry/images/annulus.svg" alt="annulus r and R" height="164" width="164" ><br>
|
||
In our case <b>R = x</b> and <b>r = x<sup>3</sup></b></p>
|
||
<p>In effect this is the <b>same as the disk method</b>, except we subtract one disk from another.</p>
|
||
<p>And so our integration looks like:</p>
|
||
<div class="center large">Volume =
|
||
<div class="intgl">
|
||
<div class="to">1</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> <span class="times">π</span> (x)<sup>2</sup> − <span class="times">π</span> (x<sup>3</sup>)<sup>2</sup> dx</div>
|
||
<!-- Volume = INT{0, 1} PI (x)^2 - PI (x^3 )^2 dx -->
|
||
<p>Have pi outside (on both functions) and simplify (x<sup>3</sup>)<sup>2</sup> = x<sup>6</sup>:</p>
|
||
<div class="center large">Volume = <span class="times">π</span>
|
||
<div class="intgl">
|
||
<div class="to">1</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> x<sup>2</sup> − x<sup>6</sup> dx</div>
|
||
<!-- Volume = PI INT{0, 1} x^2 − x^6 dx -->
|
||
<p>The integral of x<sup>2</sup> is x<sup>3</sup>/3 and the integral of x<sup>6</sup> is x<sup>7</sup>/7</p>
|
||
<p>And so, going between 0 and 1 we get:</p>
|
||
<p class="center">Volume = <span class="times">π</span> [ (1<sup>3</sup>/3 − 1<sup>7</sup>/7 ) − (0−0) ]</p>
|
||
<p class="center">≈ 0.598...</p>
|
||
</div>
|
||
<p>So the Washer method is like the Disk method, but with the inner disk subtracted from the outer disk.</p>
|
||
<p> </p>
|
||
<p> </p>
|
||
|
||
<div class="related">
|
||
<a href="solids-revolution-shells.html">Solids of Revolution by Shells</a>
|
||
<a href="index.html">Calculus Index</a>
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