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Stationary Points and Optimisation</a></label></li><li class="expanded parent selected"><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="55-58-integration-and-trapezoidal-rule.html">5.5 & 5.8 Integration and Trapezoidal rule</a></label></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> 5.5 & 5.8 Integration and Trapezoidal rule <a href="#" class="mark-page-favorite pull-right" data-pid="2197" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsapplications.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1078/calculus.html">Calculus</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">5.5 & 5.8 Integration and Trapezoidal rule</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <div class="intro-card" readonly="false"><img class="intro-image" readonly="true" src="../../images/page-pic-1.jpg"> <div class="content" readonly="true"> <p class="text">In this section of the course you will learn about an area of calculus called integration. This an be considered in multiple ways, both as the inverse of differentiation (anti-differentiation) and in finding areas under curves. We will also use the trapezoidal rule to estimate the area under a curve. </p> </div> </div> <div class="panel panel-has-colored-body panel-turquoise"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div>Key Concepts</div> </div> <div class="panel-body"> <p>In this unit you should learn to...</p> <ul> <li>Integrate polynomial functions. </li> <li>Find an original function through integration (anti-differentiation) with a boundary condition. </li> <li>Evaluate definite integrals on the GDC and hence find the area between a curve, the x-axis and an upper and lower bound. </li> <li>Estimate areas under curves using the trapezoidal rule. </li> </ul> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <p>Keep track of your progress on this page and practice the exam questions on this <a href="../../files/integration-and-trapezoidal-rule-record-sheet.pdf"><img alt="" src="../../files/integration-and-trapezoidal-rule-record-sheet.pdf">Integration and Trapezoidal Rule activity sheet</a>.</p> <h3> </h3> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>1. Introducing Integration as Anti-differentiation. </p> </div> </div> <div class="panel-body"> <div> <p>This is an introduction to integration with worked examples.<iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/537652902" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>2. Anti-differentiation with a Boundary Condition</p> </div> </div> <div class="panel-body"> <div> <p>This video works through a number of examples of finding an original function. There will be specific attention to questions which have a different notation and style, but ask exactly the same thing. <iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/534593407" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3. Definite Integration</p> </div> </div> <div class="panel-body"> <div> <p>What is the different between integration you've done previously and definite integration? How is integration connected to area?<iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/537654812" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p> 4. The Trapezoidal Rule </p> </div> </div> <div class="panel-body"> <div> <p>This video provides everything you need to know about estimating the area under a curve using the trapezoidal rule. <iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/537656654" webkitallowfullscreen="" width="100%"></iframe></p> </div> </div> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <p>These slides summarise the essential understanding and skills in this topic. </p> <div id="carousel-233" class="dynamic-gallery carousel slide" data-id="233"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-233/flash-1.png" data-fancybox="gallery-233" title="" data-caption=""><img alt="" src="../../../std-galleries/15-233/flash-1.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-233/flash-2.png" data-fancybox="gallery-233" title="" data-caption=""><img alt="" src="../../../std-galleries/15-233/flash-2.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-233/flash-3.png" data-fancybox="gallery-233" title="" data-caption=""><img alt="" src="../../../std-galleries/15-233/flash-3.png"></a></div></div><a class="left carousel-control" href="#carousel-233" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-233" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-233/flash-1-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-233/flash-2-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/15-233/flash-3-thumb128.jpg"><li></li></ol> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Click on the hidden box icon to start the quiz.</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="tib-quiz" data-stats="15-576-2197"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Evaluate <span class="math-tex">\(\int_{ }^{ }6x-x^{2}+5\ dx\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(6-2x+c\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(6x^{2}-\frac{1}{3}x^{3}+5x+c\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(3x^{2}-\frac{1}{3}x^{3}+5x\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(3x^{2}-\frac{1}{3}x^{3}+5x+c\)</span></label></p></div><div class="q-explanation"><p>See answer given. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Evaluate <span class="math-tex">\(\int_{ }^{ }t-\frac{3}{t^{4}}\ dt\)</span></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\frac{t^{2}}{2}-\frac{1}{t^{3}}+c\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{t^{2}}{2}+\frac{3}{5t^{5}}+c\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{t^{2}}{2}+\frac{1}{t^{3}}+c\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{t^{2}}{2}-\frac{3}{t^{3}}+c\)</span></span></label> </p></div><div class="q-explanation"><p>If we integrate each term separately, then <span class="math-tex">\(t\)</span> is relatively simple to integrate using the power rule; add one to the power and divide by the new power. This gives <span class="math-tex">\(\frac{t^{2}}{2}\)</span>. </p><p>For <span class="math-tex">\(-\frac{3}{t^{4}}\)</span>, we start by turning it into index form before integrating so that we can use the power rule. </p><p><span class="math-tex">\(\frac{3}{t^{4}}=3t^{-4}\)</span></p><p>Then perform the integration: <span class="math-tex">\(\int_{ }^{ }3t^{-4}\ dt=\frac{3t^{-3}}{-3}+c=-t^{-3}+c\)</span></p><p>We finalise by turning the answer back into fractional form: <span class="math-tex">\(-t^{-3}+c=-\frac{1}{t^{3}}+c\)</span></p><p>Putting this all together we obtain: <span class="math-tex">\(\frac{t^{2}}{2}-\frac{1}{t^{3}}+c\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(f'\left(x\right)=\frac{1}{2}x-4\)</span>, find <span class="math-tex">\(f(x)\)</span>.</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(f\left(x\right)=\frac{1}{4}x^{2}-4x\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(f\left(x\right)=\frac{1}{4}x^{2}-4x+c\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(f\left(x\right)=\frac{1}{2}x^{2}-4x+c\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(f\left(x\right)=\frac{1}{2}\)</span></span></label> </p></div><div class="q-explanation"><p>Integrating the derivative gives the original function. See the answer given. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Evaluate <span class="math-tex">\(\int_{ }^{ }\left(x+4\right)^{2}-2\ dx\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{x^{3}}{3}+4x^{2}+12x+c\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{x^{3}}{3}+8x^{2}+14x+c\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{x^{3}}{3}+4x^{2}+c\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\frac{x^{3}}{3}+4x^{2}+14x+c\)</span></span></label> </p></div><div class="q-explanation"><p>Before integrating, expand the bracket and simplify: </p><p><span class="math-tex">\(\left(x+4\right)^{2}-2=\left(x+4\right)\left(x+4\right)-2=x^{2}+8x+16-2=x^{2}+8x+14\)</span></p><p>Hence, <span class="math-tex">\(\int_{ }^{ }x^{2}+8x+14\ dx=\frac{x^{3}}{3}+4x^{2}+14x+c\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Find <span class="math-tex">\(g\left(x\right)\)</span> given that <span class="math-tex">\(g'\left(x\right)=5-4x^{2}\)</span> and <span class="math-tex">\(g\left(3\right)=-1\)</span>.</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(g\left(x\right)=5x-\frac{4x^{3}}{3}+c\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(g\left(x\right)=5x-\frac{4x^{3}}{3}+6\frac{2}{3}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(g\left(x\right)=5x-\frac{4x^{3}}{3}+20\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(g\left(x\right)=-8x+23\)</span></span></label> </p></div><div class="q-explanation"><p>First integrate <span class="math-tex">\(g'\left(x\right)\)</span> to obtain <span class="math-tex">\(g\left(x\right)=5x-\frac{4x^{3}}{3}+c\)</span>. </p><p>Then substitute the boundary condition values into <span class="math-tex">\(g\left(x\right)\)</span>to obtain the answer shown. </p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(f'\left(x\right)=mx+4\)</span>, where <span class="math-tex">\(m\)</span> is a constant. If the points <span class="math-tex">\(\left(0,3\right)\)</span> and <span class="math-tex">\(\left(-2,7\right)\)</span> lie on <span class="math-tex">\(f\left(x\right)\)</span>, find the value of <span class="math-tex">\(m.\)</span></p></div><div class="q-answer"><p>m = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span> </p></div><div class="q-explanation"><p>Start by integrating <span class="math-tex">\(f'\left(x\right)\)</span> to obtain <span class="math-tex">\(f\left(x\right)=\frac{mx^{2}}{2}+4x+c\)</span>.</p><p>Notice that we have two boundary conditions (i.e. two separate coordinates) and two unknowns in the equation, <span class="math-tex">\(m\)</span> and <span class="math-tex">\(c\)</span>. Substitute each coordinate into <span class="math-tex">\(f\left(x\right)\)</span> to find each of these unknowns. </p><p>Substituting <span class="math-tex">\(\left(0,3\right)\)</span> into <span class="math-tex">\(f\left(x\right)\)</span> gives <span class="math-tex">\(f\left(x\right)=\frac{mx^{2}}{2\ }+4x+3\)</span>. </p><p>Then substituting <span class="math-tex">\(\left(-2,7\right)\)</span> gives <span class="math-tex">\(m=1\)</span>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Evaluate <span class="math-tex">\(\int_{2}^{4}x^{2}-\frac{1}{x^{2}}dx\)</span></p></div><div class="q-answer"><p>Answer = <input type="text" style="height: auto;" data-c="18.9"> <span class="review"></span> </p></div><div class="q-explanation"><p>Input this definite integral into the GDC to obtain <span class="math-tex">\(18.9\)</span> (<span class="math-tex">\(3\)</span> s.f.)</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Use the trapezoidal rule to find an estimate of the area between the curve, <span class="math-tex">\(f\left(x\right)=\sqrt{x}+1\)</span>, and the x-axis for <span class="math-tex">\(0\le x\le3\)</span> with <span class="math-tex">\(n=2\)</span>.</p></div><div class="q-answer"><p>Estimated area = <input type="text" style="height: auto;" data-c="6.14"> <span class="review"></span> </p></div><div class="q-explanation"><p>Step 1: <span class="math-tex">\(h=\frac{\left(3-0\right)}{2}=1.5\)</span></p><p>Step 2:</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>x</td><td>0</td><td>1.5</td><td>3</td></tr><tr><td>y</td><td>1</td><td>2.2247...</td><td>2.7320...</td></tr></tbody></table><p>Step 3:</p><p>Area = <span class="math-tex">\(\frac{1}{2}\cdot1.5\left(1+2\left(2.2247...\right)+2.7320...\right)=6.14\ unit^{2}\ \left(3.s.f\right)\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Estimate the area between the x-axis and the curve <span class="math-tex">\(y=f\left(x\right)\)</span> using the coordinates for <span class="math-tex">\(0\le x\le4\)</span>.</p><p><img alt="" src="../../images/picture1(9).png" style="width: 200px; height: 361px;"></p></div><div class="q-answer"><p>Estimated area = <input type="text" style="height: auto;" data-c="23"> <span class="review"></span> </p></div><div class="q-explanation"><p>The coordinates in table form are: </p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td>x</td><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td>y</td><td>3</td><td>3.5</td><td>5</td><td>7.5</td><td>11</td></tr></tbody></table><p>Hence, utilising the trapezoidal rule: </p><p><span class="math-tex">\(\frac{1}{2}\cdot1\left(3+2\left(3.5+5+7.5\right)+11\right)=23\ units^{2}\)</span></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Complete the sentence: Using the trapezoidal rule, the estimated area gets closer to the actual area between a curve and the x-axis when...</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> ...you decrease the number of trapezoids, <span class="math-tex">\(n\)</span></label>, under the curve. </p><p><label class="radio"><input type="radio"> ...you change the lower bound and the upper bound given.</label></p><p><label class="radio"><input class="c" type="radio"> ...you increase the number of trapezoids, <span class="math-tex">\(n\)</span></label>, under the curve. </p></div><div class="q-explanation"><p>See answer given.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </section> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam Style Questions</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <p><img alt="" src="../../images/examq1(1).png" style="width: 700px; height: 101px;"></p> <p>Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/544320688" webkitallowfullscreen="" width="100%"></iframe></section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <p><img alt="" src="../../images/examq2(1).png" style="width: 700px; height: 133px;"></p> <p>Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/544321897" webkitallowfullscreen="" width="100%"></iframe></section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <p><img alt="" src="../../images/examq3(1).png" style="width: 700px; height: 261px;"></p> <p>Video solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><iframe allowfullscreen="" frameborder="0" height="auto" mozallowfullscreen="" src="https://player.vimeo.com/video/544323123" webkitallowfullscreen="" width="100%"></iframe></section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-red"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Just for Fun</p> </div> </div> <div class="panel-body"> <div> <p>Have you ever wondered about the connection between the area and circumference of a circle?</p> <p>If you integrate the circumference you obtain the area:</p> <p style="text-align: center;"><span class="math-tex">\(\int_{0}^{r}2\pi r\ dr\ =\ \pi r^{2}\)</span></p> <p>Why?</p> <p>The integral sign actually stands for "infinite sum". That's the reason why it looks like an elongated "S" shape. So the the equation above means the following:</p> <p style="text-align: center;"><em>If you draw an infinite number of smaller circles inside a circle, then sum all of the infinite circumferences, then those infinite lines added up give you an area (the more lengths you add, the closer you'll get to the area). </em></p> <p style="text-align: center;"><em><img alt="" src="../../images/picture1(11).png" style="width: 600px; height: 100px;"></em></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="2197"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>5.5 & 5.8 Integration and Trapezoidal rule</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" 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