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      <text>14.1.1.1 The Limit of a Sum</text>
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      <text><![CDATA[<h3> The Limit of a Sum </h3>
<p></p><center><img src="@@PLUGINFILE@@/Figure%201.svg" alt="Graph showing the area under the function y(x) betwwen the points x=a and x=b." width="240" height="135" class="img-responsive atto_image_button_text-bottom"><br></center><p></p>

<p>Consider the graph of the positive function <span class="math inline">\(y(x)\)</span> shown in the figure above. Suppose we are interested in finding the area under the graph between <span class="math inline">\(x=a\)</span> and <span class="math inline">\(x=b\)</span>. One way in which this area can be <span><strong>approximated</strong></span> is to divide it into a number of rectangles of equal width, find the area of each rectangle, and then add up all these individual rectangular areas. This is illustrated in the figure below on the left, which shows the area divided into <span class="math inline">\(n\)</span> rectangles (with some small discrepancies at the tops), and the figure below on the right which shows the dimensions of a typical rectangle which is located at <span class="math inline">\(x=x_k\)</span>.</p>

<p style="text-align: center;"><img src="@@PLUGINFILE@@/Figure%202b.svg" alt="As described in the text above." width="620" height="163" class="img-responsive atto_image_button_text-bottom"><br></p>

<p>We wish to find an expression for the area under a curve based on the sum of many rectangles. Firstly, we note that the distance from <span class="math inline">\(x=a\)</span> to <span class="math inline">\(x=b\)</span> is <span class="math inline">\(b-a\)</span>. In Figure 2a the area has been divided into <span class="math inline">\(n\)</span> rectangles. If <span class="math inline">\(n\)</span> rectangles span the distance from <span class="math inline">\(a\)</span> to <span class="math inline">\(b\)</span> the width of each rectangle is <span class="math inline">\(\dfrac{b-a}{n}\)</span></p>

<p>It is conventional to label the width of each rectangle as <span class="math inline">\(\delta x\)</span>, i.e. <span class="math inline">\(\delta x =
\dfrac{b-a}{n}\)</span>. We label the <span class="math inline">\(x\)</span> coordinates at the left-hand side of the rectangles as <span class="math inline">\(x_1\)</span>, <span class="math inline">\(x_2\)</span> up to <span class="math inline">\(x_n\)</span> (here <span class="math inline">\(x_1=a\)</span> and <span class="math inline">\(x_{n+1}=b\)</span>). A typical rectangle, the <span class="math inline">\(k\)</span>th rectangle, is shown in the figure above on the right. Note that its height is <span class="math inline">\(y(x_k)\)</span>, so its area is <span class="math inline">\(y(x_k)\times \delta x\)</span>.</p>

<p> The sum of the areas of all \(n\) rectangles is then \[y(x_1)\delta x + y(x_2)\delta x + y(x_3)\delta x + \dots  + y(x_n)\delta x\] which we write concisely using sigma notation as \[\sum_{k=1}^n y(x_k)\delta x\] </p>

<p>This quantity gives us an estimate of the area under the curve but it is not exact. To improve the estimate we must take a large number of very thin rectangles. So, what we want to find is the value of this sum when <span class="math inline">\(n\)</span> tends to infinity and <span class="math inline">\(\delta x\)</span> tends to zero. We write this value as \[ \lim_{n\rightarrow\infty} \sum_{k=1}^n y(x_k)\delta x\]</p>

<p>The lower and upper limits on the sum correspond to the first rectangle and last rectangle where <span class="math inline">\(x=a\)</span> and <span class="math inline">\(x=b\)</span> respectively and so we can write this limit in the equivalent form \[ \displaystyle{\lim_{\delta x\rightarrow 0}\sum_{x=a}^{x=b}y(x)\delta x}.\]</p>

<p>Here, as the number of rectangles increases without bound we drop the subscript <span class="math inline">\(k\)</span> from <span class="math inline">\(x_k\)</span> and write <span class="math inline">\(y(x)\)</span> which is the value of <span class="math inline">\(y\)</span> at a ‘typical’ value of <span class="math inline">\(x\)</span>. If this sum can actually be found, it is called the definite integral of <span class="math inline">\(y(x)\)</span>, from <span class="math inline">\(x=a\)</span> to <span class="math inline">\(x=b\)</span> and it is written <span class="math inline">\(\displaystyle{\int_a^by(x) \, \mathrm{d}x}\)</span>.</p>
<p> Therefore we have the following definition: </p>

<div class="HELM_keypoint">
<h4>Key Point</h4>
<p>The definite integral \(\displaystyle{ \int_a^b y(x) \, \mathrm{d}x }\) is defined as \(\displaystyle{ \lim_{\delta x\rightarrow 0}\sum_{x=a}^{x=b}y(x)\delta x }\).</p>
</div>
<p></p>

<p>Note that the quantity <span class="math inline">\(\delta x\)</span> represents the thickness of a small but finite rectangle. When we have taken the limit as <span class="math inline">\(\delta x\)</span> tends to zero to obtain the integral, we write <span class="math inline">\(\mathrm{d}x\)</span>, which reminds us of the variable of integration.</p>

<p>This process of dividing an area into very small regions, performing a calculation on each region, and then adding the results by means of an integral is very important. This will become apparent when finding volumes, centres of gravity, moments of inertia etc.</p>

<hr>
<h4 class="HELM_example">Example</h4>
<p>The area under the graph of <span class="math inline">\(y=x^2\)</span> between <span class="math inline">\(x=0\)</span> and <span class="math inline">\(x=1\)</span> is to be found by approximating it by a large number of thin rectangles and finding the limit of the sum of their areas. This is equivalent to \(\displaystyle{\lim_{\delta x\rightarrow 0}\sum_{x=0}^{x=1}y(x)\,\delta x}\). Write down the integral which this sum defines and evaluate it to obtain the area under the curve.</p>

<h4 class="HELM_solution">Solution</h4>
<p>The limit of the sum defines the integral \(\displaystyle{\int_0^1y(x) \, \mathrm{d}x}\). Here \(y=x^2\) and so \( \displaystyle{\int_0^1 x^2 \, \mathrm{d}x = \left[\frac{x^3}{3}\right]_0^1=\frac{1}{3}}\).</p>
<hr>

<p>To show that the process of taking the limit of a sum actually works we investigate the problem in detail.
We use the idea of the limit of a sum to find the area under the graph of \(y=x^2\) between \(x=0\) and \(x=1\), as illustrated in the figure below.</p>
<p></p><center><img src="@@PLUGINFILE@@/Figure%203.svg" alt="Graph displaying the curve y=x^2 between 0 and 1, split into n rectangles." width="242" height="148" class="img-responsive atto_image_button_text-bottom"><br></center><p></p>]]></text>
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