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    <name>
      <text>14.4.1a Length of curve 1</text>
    </name>
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      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Find the length of the curve \(y = \cosh x\) between \(x = 0\) and \(x = 2\) shown in the diagram.</p>
<p style="text-align: center;"><img src="@@PLUGINFILE@@/14-4-Task-cosh.svg" alt="Graph of y=cosh(x)  between x=0 and x=2." width="300" height="250" class="img-responsive atto_image_button_text-bottom"><br></p>
<p>First write down \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\)</p>
<p>\(\dfrac{\mathrm{d}y}{\mathrm{d}x}=\) [[input:ans1]] [[validation:ans1]]</p>
<p>[[feedback:prt1]]</p>
<p>Hence write down the required integral:</p>
<p>\(\displaystyle{\int_0^2}\)[[input:ans2]]&nbsp; \(\mathrm{d}x\) [[validation:ans2]]</p>
<p>[[feedback:prt2]]</p>
<p>This integral can be evaluated by making use of the hyperbolic identity \(\cosh^2 x - \sinh^2 x \equiv 1\).</p>
<p>Write down the integral which results after applying this identity:</p>
<p>\(\displaystyle{\int_0^2}\)[[input:ans3]] \(\mathrm{d}x\)[[validation:ans3]]</p>
<p>[[feedback:prt3]]</p>
<p>Perform the integration to find the required length, giving your answer to <b>2 decimal places</b>.</p>
<p>[[input:ans4]] [[validation:ans4]]</p>
<p>[[feedback:prt4]]</p>]]></text>
<file name="14-4-Task-cosh.svg" path="/" encoding="base64">[encoded data, 30520 characters]</file>
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    <generalfeedback format="html">
      <text><![CDATA[<p>The formula in the Key Point tells us that, given a curve with equation \(y=f(x)\), the length of the curve between the points where \(x=a\) and \(x=b\) is given by the formula \[\int_a^b \sqrt{1+\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2} \mathrm{d}x\]</p>
<p>If \(y=\cosh x\) then \(\frac{\mathrm{d}y}{\mathrm{d}x}=\sinh x\), therefore the required integral is \[\int_0^2 \sqrt{1+\sinh^2 x} \mathrm{d}x\]</p>
<p>Making use of the hyperbolic identity \(\cosh^2 x - \sinh^2 x \equiv 1\), the integral can be written as \[\int_0^2 \cosh x \mathrm{d}x\]</p>
<p>Therefore, the length of the curve \(y=\cosh x\) between \(x=0\) and \(x=2\) is \[\big[\sinh x\big]^2_0=\sinh(2)-\sinh(0)=3.62686=3.63 \]</p>]]></text>
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    <stackversion>
      <text>2020120600</text>
    </stackversion>
    <questionvariables>
      <text>cur: cosh(x);
ta1: diff(cur,x);
ta2: sqrt(1+sinh(x)^2);
ta3: cosh(x);
len: integrate(cosh(x), x, 0, 2);
ta4: decimalplaces(float(len),2);</text>
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      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
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