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1025 lines
18 KiB
HTML
1025 lines
18 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Advanced.dwt" --><!-- DW6 -->
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<title>Arc Length (Calculus)</title>
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<div id="adTop"></div>
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<div id="hdr"></div>
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<div id="logo"><a href="../index.html"><img src="../images/style/logo-adv.svg" alt="Math is Fun"></a></div>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Arc Length</h1>
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<p class="center"><b>Using Calculus to find the length of a curve</b>.<br>
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<i>(Please read about <a href="derivatives-introduction.html">Derivatives</a> and <a href="integration-introduction.html"> Integrals</a> first)
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</i></p>
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<p>Imagine we want to find the length of a curve between two points. And the curve is smooth (the derivative is <a href="continuity.html">continuous</a>).</p>
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<p class="center"><img src="images/arc-length-2.svg" alt="arc length curve"></p>
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<p>First we break the curve into small lengths and use the <a href="../algebra/distance-2-points.html">Distance Between 2 Points</a> formula on each length to come up with an approximate answer:</p>
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<p class="centerfull"><img src="images/arc-length-1.gif" alt="arc length between points" height="239" width="360"></p>
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<p>The distance from <b>x<sub>0</sub></b> to <b>x<sub>1</sub></b> is:</p>
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<p class="center large">S<sub>1</sub> = <span style="font-size: 120%;">√</span><span class="overline"> (x<sub>1</sub> − x<sub>0</sub>)<sup>2</sup> + (y<sub>1</sub> − y<sub>0</sub>)<sup>2</sup></span></p>
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<p>And let's use <b> Δ</b> (delta) to mean the difference between values, so it becomes:</p>
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<p class="center large">S<sub>1</sub> = <span style="font-size: 120%;">√</span><span class="overline"> <span class="overline">(Δx<sub>1</sub>)<sup>2</sup> + (Δy<sub>1</sub>)<sup>2</sup></span></span></p>
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<p>Now we just need lots more:</p>
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<p class="center large">S<sub>2</sub> = <span style="font-size: 120%;">√</span><span class="overline">(Δx<sub>2</sub>)<sup>2</sup> + (Δy<sub>2</sub>)<sup>2</sup></span><span style="border-top:1px solid; "></span><br>
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S<sub>3</sub> = <span style="font-size: 120%;">√</span><span class="overline">(Δx<sub>3</sub>)<sup>2</sup> + (Δy<sub>3</sub>)<sup>2</sup></span><span style="border-top:1px solid; "></span><br>
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...<br>
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...<br>
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S<sub>n</sub> = <span style="font-size: 120%;">√</span><span class="overline">(Δx<sub>n</sub>)<sup>2</sup> + (Δy<sub>n</sub>)<sup>2</sup></span><span class="overline"></span></p>
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<p> </p>
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<p>We can write all those many lines in just <b>one line</b> using a <a href="../algebra/sigma-notation.html">Sum</a>:</p>
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<!-- s APR SIG{i=1, n} SQR (DELx_i~)^2 + (DELy_i~)^2 -->
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<div class="center large">
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S ≈
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<div class="sigma">
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<div class="to">n</div>
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<div class="symb"></div>
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<div class="from">i=1</div>
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</div>
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<!-- SIG{i=1, n} -->
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<span style="font-size: 120%;">√</span><span class="overline">(Δx<sub>i</sub>)<sup>2</sup> + (Δy<sub>i</sub>)<sup>2</sup></span>
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</div>
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<p>But we are still doomed to a large number of calculations!</p>
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<p>Maybe we can make a big spreadsheet, or write a program to do the calculations ... but lets try something else.</p>
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<p>We have a cunning plan:</p>
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<ul>
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<li>have all the <b>Δx<sub>i</sub></b> be <b>the same</b> so we can extract them from inside the square root</li>
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<li>and then turn the sum into an integral.</li>
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</ul>
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<p>Let's go:</p>
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<p>First, divide <i>and</i> multiply <b>Δy<sub>i</sub></b> by <b>Δx<sub>i</sub></b>:</p>
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<div class="center large">
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S ≈
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<div class="sigma">
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<div class="to">n</div>
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<div class="symb"></div>
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<div class="from">i=1</div>
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</div>
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<!-- SIG{i=1, n} -->
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<span style="font-size: 120%;">√</span><span class="overline">(Δx<sub>i</sub>)<sup>2</sup> + (Δx<sub>i</sub>)<sup>2</sup>(Δy<sub>i</sub>/Δx<sub>i</sub>)<sup>2</sup></span>
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</div>
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<p>Now factor out <b>(Δx<sub>i</sub>)<sup>2</sup></b>:</p>
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<div class="center large">
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S ≈
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<div class="sigma">
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<div class="to">n</div>
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<div class="symb"></div>
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<div class="from">i=1</div>
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</div>
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<!-- SIG{i=1, n} -->
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<span style="font-size: 120%;">√</span><span class="overline">(Δx<sub>i</sub>)<sup>2</sup>(1 + (Δy<sub>i</sub>/Δx<sub>i</sub>)<sup>2</sup>)</span>
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</div>
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<p>Take <b>(Δx<sub>i</sub>)<sup>2</sup></b> out of the square root:</p>
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<div class="center large">
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S ≈
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<div class="sigma">
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<div class="to">n</div>
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<div class="symb"></div>
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<div class="from">i=1</div>
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</div>
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<!-- SIG{i=1, n} -->
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<span style="font-size: 120%;">√</span><span class="overline">1 + (Δy<sub>i</sub>/Δx<sub>i</sub>)<sup>2</sup></span> Δx<sub>i</sub>
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</div>
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<p>Now, as <b>n approaches infinity</b> (as we head towards an infinite number of slices, and each slice gets smaller) we get:</p>
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<div class="center large">
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S =
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<div style="display:inline-block; text-align:center; transform: translateY(25%);">
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||
<div style=" font-size:130%; font-family: 'Times New Roman', Times, serif; transform: translateY(35%); ">lim</div>
|
||
<div style="font-family: 'Times New Roman', Times, serif; ">n→∞</div>
|
||
</div>
|
||
|
||
<div class="sigma">
|
||
<div class="to">n</div>
|
||
<div class="symb"></div>
|
||
<div class="from">i=1</div>
|
||
</div>
|
||
<!-- SIG{i=1, n} -->
|
||
<span style="font-size: 120%;">√</span><span class="overline">1 + (Δy<sub>i</sub>/Δx<sub>i</sub>)<sup>2</sup></span> Δx<sub>i</sub>
|
||
</div>
|
||
|
||
|
||
|
||
<p>We now have an <a href="integration-introduction.html">integral</a> and we write <b>dx</b> to mean the <b>Δx</b> slices are approaching zero in width (likewise for <b>dy)</b>:</p>
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<div class="center large">S =
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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>
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<span style="font-size: 120%;">√</span><span class="overline">1+(dy/dx)<sup>2</sup></span> dx
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</div>
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<p>And <b>dy/dx</b> is the <a href="derivatives-introduction.html">derivative</a> of the function f(x), which can also be written<b> f’(x)</b>:</p>
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<div class="def">
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||
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<div class="center larger">S =
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<div class="intgl">
|
||
<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>
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<span style="font-size: 120%;">√</span><span class="overline">1+(f’(x))<sup>2</sup></span> dx<br>
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<b>The Arc Length Formula</b>
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</div>
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||
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||
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</div>
|
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<p>And now suddenly we are in a much better place, we don't need to add up lots of slices, we can calculate an exact answer (if we can solve the differential and integral).</p>
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<p><i>Note: the integral also works with respect to y, useful if we happen to know x=g(y):</i></p>
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||
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<div class="center large">S =
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<div class="intgl">
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||
<div class="to">d</div>
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<div class="symb"></div>
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<div class="from">c</div>
|
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</div> <span style="font-size: 120%;">√</span><span class="overline">1+(g’(y))<sup>2</sup></span> dy
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</div>
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||
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<p>So our steps are:</p>
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<ul>
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<li>Find the derivative of <b>f’(x)</b></li>
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||
<li>Solve the integral of <b><span style="font-size: 120%;">√</span><span style="border-top:2px solid #009; ">1 + (f’(x))<sup>2</sup></span> dx</b></li>
|
||
</ul>
|
||
<p>Some simple examples to begin with:</p>
|
||
<div class="example">
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/arc-length-4.gif" alt="arc length constant" height="105" width="180"></p>
|
||
<h3>Example: Find the length of f(x) = 2 between x=2 and x=3</h3>
|
||
<p>f(x) is just a horizontal line, so its derivative is <b>f’(x) = 0</b></p>
|
||
<p>Start with:</p>
|
||
<div class="center large">S =
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="intgl">
|
||
<div class="to">3</div>
|
||
<div class="symb"></div>
|
||
<div class="from">2</div>
|
||
</div> <span style="font-size: 120%;">√</span><span class="overline">1+(f’(x))<sup>2</sup></span> dx
|
||
</div>
|
||
<p>Put in <b>f’(x) = 0</b>:</p>
|
||
<div class="center large">S =
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="intgl">
|
||
<div class="to">3</div>
|
||
<div class="symb"></div>
|
||
<div class="from">2</div>
|
||
</div> <span style="font-size: 120%;">√</span><span class="overline">1+0<sup>2</sup></span> dx
|
||
</div>
|
||
|
||
<p>Simplify:</p>
|
||
<div class="center large">S =
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="intgl">
|
||
<div class="to">3</div>
|
||
<div class="symb"></div>
|
||
<div class="from">2</div>
|
||
</div> dx</div>
|
||
|
||
<p>Calculate the Integral:</p>
|
||
<div class="center larger">S = 3 − 2 = 1<br>
|
||
</div>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<p><br></p>
|
||
<p>So the arc length between 2 and 3 is 1. Well of course it is, but it's nice that we came up with the right answer!</p>
|
||
<p>Interesting point: the "(1 + ...)" part of the Arc Length Formula guarantees we get <b>at least</b> the distance between x values, such as this case where <b> f’(x)</b> is zero.</p>
|
||
</div>
|
||
<div class="example">
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/arc-length-5.gif" alt="arc length slope" height="106" width="180"></p>
|
||
<h3>Example: Find the length of f(x) = x between x=2 and x=3</h3>
|
||
<p>The derivative <b>f’(x) = 1</b></p>
|
||
<p><br>
|
||
Start with:</p>
|
||
|
||
<div class="center large">S =
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="intgl">
|
||
<div class="to">3</div>
|
||
<div class="symb"></div>
|
||
<div class="from">2</div>
|
||
</div> <span style="font-size: 120%;">√</span><span class="overline">1+(f’(x))<sup>2</sup></span> dx
|
||
</div><br>
|
||
<p>Put in <b>f’(x) = 1</b>:</p>
|
||
<div class="center large">S =
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="intgl">
|
||
<div class="to">3</div>
|
||
<div class="symb"></div>
|
||
<div class="from">2</div>
|
||
</div> <span style="font-size: 120%;">√</span><span class="overline">1+(1)<sup>2</sup></span> dx
|
||
</div>
|
||
|
||
<p>Simplify:</p>
|
||
<div class="center large">S =
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="intgl">
|
||
<div class="to">3</div>
|
||
<div class="symb"></div>
|
||
<div class="from">2</div>
|
||
</div> <span style="font-size: 120%;">√</span><span class="overline">2</span> dx
|
||
</div>
|
||
|
||
<p>Calculate the Integral:</p>
|
||
<div class="larger center">
|
||
S = <span style="border-top:1px solid; "></span>(3−2)<span style="font-size: 120%;">√</span><span class="overline">2</span> = <span style="font-size: 120%;">√</span><span class="overline">2</span>
|
||
<div><br>
|
||
</div></div>
|
||
|
||
<p>And the diagonal across a unit square really is the square root of 2, right?</p>
|
||
</div>
|
||
<p>OK, now for the harder stuff. A real world example.</p>
|
||
<div class="example">
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/rope-bridge.jpg" alt="rope bridge" height="226" width="300"></p>
|
||
<h3>Example: Metal posts have been installed <b>6m apart</b> across a gorge.<br>
|
||
<br>
|
||
Find the length for the hanging bridge that follows the curve:</h3>
|
||
<p class="center large">f(x) = 5 cosh(x/5)</p>
|
||
<div style="clear:both"></div>
|
||
<p>Here is the actual curve:</p>
|
||
<p class="center"><img src="images/catenary-1.gif" alt="catenary graph" height="131" width="480"></p>
|
||
<p>Let us solve the general case first!</p>
|
||
<p>A hanging cable forms a curve called a <b>catenary</b>:</p>
|
||
<p class="center large">f(x) = a cosh(x/a)</p>
|
||
<p class="center">Larger values of <b>a</b> have less sag in the middle<br>
|
||
And "cosh" is the <a href="../sets/function-hyperbolic.html">hyperbolic cosine</a> function.</p>
|
||
<p>The derivative is <b>f’(x) = sinh(x/a)</b></p>
|
||
<p>The curve is symmetrical, so it is easier to work on just half of the catenary, from the center to an end at "b":</p>
|
||
<p><br></p>
|
||
<p>Start with:</p>
|
||
<div class="center large">S =
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="intgl">
|
||
<div class="to">b</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> <span style="font-size: 120%;">√</span><span class="overline">1+(f’(x))<sup>2</sup></span> dx
|
||
</div>
|
||
|
||
<p>Put in <b>f’(x) = sinh(x/a)</b>:</p>
|
||
<div class="center large">S =
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="intgl">
|
||
<div class="to">b</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> <span style="font-size: 120%;">√</span><span class="overline">1 + sinh<sup>2</sup>(x/a)</span> dx
|
||
</div>
|
||
|
||
<p>Use the identity <b>1 + sinh<sup>2</sup>(x/a) = cosh<sup>2</sup>(x/a):</b></p>
|
||
|
||
<div class="center large">S =
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="intgl">
|
||
<div class="to">b</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> <span style="font-size: 120%;">√</span><span class="overline">cosh<sup>2</sup>(x/a)</span> dx
|
||
</div>
|
||
|
||
|
||
<p>Simplify:</p>
|
||
<div class="center large">S =
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="intgl">
|
||
<div class="to">b</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> cosh(x/a) dx
|
||
</div>
|
||
|
||
|
||
<p>Calculate the Integral:</p>
|
||
<p class="center large">S = a sinh(b/a)</p>
|
||
|
||
|
||
|
||
<p>Now, remembering the symmetry, let's go from −b to +b:</p>
|
||
<p class="center larger">S = 2a sinh(b/a)</p>
|
||
|
||
<p><br></p>
|
||
<p>In our <b>specific case</b> a=5 and the 6m span goes from −3 to +3</p>
|
||
<p class="center"><span class="large">S = 2×5 sinh(3/5)<br>
|
||
= <b>6.367 m</b></span> (to nearest mm)</p>
|
||
<p>This is important to know! If we build it exactly 6m in length there is <b>no way</b> we could pull it hard enough for it to meet the posts. But at 6.367m it will work nicely.</p>
|
||
|
||
</div>
|
||
<p> </p>
|
||
<div class="example">
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/arc-length-6.gif" alt="arc length graph" height="201" width="120"></p>
|
||
<h3>Example: Find the length of y = x<sup>(3/2)</sup> from x = 0 to x = 4.</h3>
|
||
<p> </p>
|
||
<p>The derivative is <b>y’ = (3/2)x<sup>(1/2)</sup></b></p>
|
||
|
||
|
||
<p><br></p>
|
||
<p>Start with:</p>
|
||
<div class="center large">S =
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="intgl">
|
||
<div class="to">4</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> <span style="font-size: 120%;">√</span><span class="overline">1+(f’(x))<sup>2</sup></span> dx
|
||
</div>
|
||
|
||
|
||
<p>Put in <b>(3/2)x<sup>(1/2)</sup></b>:</p>
|
||
<div class="center large">S =
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="intgl">
|
||
<div class="to">4</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> <span style="font-size: 120%;">√</span><span class="overline">1+((3/2)x<sup>(1/2)</sup>)<sup>2</sup></span> dx
|
||
</div>
|
||
|
||
|
||
<p>Simplify:</p>
|
||
<div class="center large">S =
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="intgl">
|
||
<div class="to">4</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> <span style="font-size: 120%;">√</span><span class="overline">1+(9/4)x</span> dx
|
||
</div>
|
||
|
||
|
||
<p>We can use <a href="integration-by-substitution.html">integration by substitution</a>:</p>
|
||
<ul>
|
||
<li>u = 1 + (9/4)x</li>
|
||
<li>du = (9/4)dx</li>
|
||
<li>(4/9)du = dx</li>
|
||
<li>Bounds: u(0)=1 and u(4)=10</li></ul>
|
||
|
||
<p>And we get:</p>
|
||
<div class="center large">S =
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="intgl">
|
||
<div class="to">10</div>
|
||
<div class="symb"></div>
|
||
<div class="from">1</div>
|
||
</div> (4/9)<span style="font-size: 120%;">√</span><span class="overline">u</span> du
|
||
</div>
|
||
|
||
<p>Integrate:</p>
|
||
<p class="center large">S = (8/27) u<sup>(3/2)</sup> from 1 to 10</p>
|
||
<p>Calculate:</p>
|
||
<p class="center large">S = (8/27) (10<sup>(3/2)</sup> − 1<sup>(3/2)</sup>) = <b>9.073...</b></p>
|
||
|
||
|
||
|
||
</div>
|
||
<h2>Conclusion</h2>
|
||
<p>The Arc Length Formula for a function f(x) is:</p>
|
||
<div class="center large">S =
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="intgl">
|
||
<div class="to">b</div>
|
||
<div class="symb"></div>
|
||
<div class="from">a</div>
|
||
</div> <span style="font-size: 120%;">√</span><span class="overline">1+(f’(x))<sup>2</sup></span> dx
|
||
</div>
|
||
|
||
|
||
<p>Steps:</p>
|
||
<ul>
|
||
<li>Take derivative of f(x)</li>
|
||
<li>Write Arc Length Formula</li>
|
||
<li>Simplify and solve integral</li>
|
||
</ul>
|
||
<p> </p>
|
||
|
||
<div class="questions">
|
||
<script>getQ(11382, 11383, 11384, 11385, 11386);</script>
|
||
</div>
|
||
|
||
<div class="related">
|
||
<a href="integration-introduction.html">Integrals</a>
|
||
<a href="derivatives-introduction.html">Derivatives</a>
|
||
<a href="derivatives-rules.html">Derivative Rules</a>
|
||
<a href="index.html">Calculus Index</a>
|
||
</div>
|
||
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|
||
|
||
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|
||
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||
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