{"id":1988,"date":"2021-08-19T16:07:00","date_gmt":"2021-08-19T16:07:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-numerical-integration\/"},"modified":"2021-11-19T03:08:19","modified_gmt":"2021-11-19T03:08:19","slug":"skills-review-for-numerical-integration","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/skills-review-for-numerical-integration\/","title":{"raw":"Skills Review for Numerical Integration","rendered":"Skills Review for Numerical Integration"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Complete a table values with solutions to an equation<\/li>\r\n \t<li>Use sigma (summation) notation to calculate sums and powers of integers<\/li>\r\n \t<li>Find the area of a trapezoid<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Numerical Integration section, we will estimate the value of definite integrals using a variety of estimation rules. Here we will review some essential topics that will help us to better understand the various estimation rules.\r\n<h2>Complete a Table of Function Values<\/h2>\r\n<div>\r\n\r\nA table of values can be used to organize the\u00a0<em>y<\/em>-values or function values that result from plugging specific\u00a0<em>x<\/em>-values into a function's equation.\r\n\r\nSuppose we want to determine the various values of the equation [latex]f(x)=2x - 1[\/latex] at certain values of <em>x<\/em>. We can begin by substituting a value for <em>x<\/em> into the equation and determining the resulting value of the function. The table below\u00a0lists some values of <em>x<\/em> from \u20133 to 3 and the resulting function values.\r\n<table style=\"width: 411px; height: 84px;\" summary=\"This is a table with 8 rows and 3 columns. The first row has columns labeled: x, y = 2x-1, (x, y). The entries in the second row are: negative 3; y = 2 times negative 3 minus 1 = negative 7; (-3, -7). The entries in the third row are: negative 2; y = 2 times negative 2 minus 1 = negative 5; (-2, -5). The entries in the fourth row are: negative1; y = 2 times negative 1 minus 1 = negative 3; (-1, -3). The entries in the fifth row are: 0; y = 2 times 0 minus 1 = negative 1; (0, -1). The entries in the sixth row are: 1; y = 2 times 1 minus 1 = 1; (1, 1). The entries in the seventh row are: 2; y = 2 times 2 minus 1 = 3; (2, 3). The entries in the eight row are: 3, y = 2 times 3 minus 1 = 5; (3,5)\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 116.5px; height: 12px;\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(x)=2x - 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 116.5px; height: 12px;\">[latex]-3[\/latex]<\/td>\r\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(-3)=2\\left(-3\\right)-1=-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 116.5px; height: 12px;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(-2)=2\\left(-2\\right)-1=-5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 116.5px; height: 12px;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(-1)=2\\left(-1\\right)-1=-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 116.5px; height: 12px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(0)=2\\left(0\\right)-1=-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 116.5px; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(2)=2\\left(2\\right)-1=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 116.5px; height: 12px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(3)=2\\left(3\\right)-1=5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can look for trends among our function values by looking at the table. For example, in this case, as\u00a0<em>x<\/em>-values increase, so do the\u00a0<em>y<\/em>-values. Also, it seems reasonable to assume, based on the table, an\u00a0<em>x<\/em>-value of 1 would result in a function value of 1. You can verify this for yourself by plugging 1 into [latex]f(x)=2x - 1[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Completing a table of function values<\/h3>\r\nCreate a table of function values for [latex]f(x)=-x+2[\/latex]. Use various integers\u00a0from -5 to 5 as the\u00a0<em>x<\/em>-values you plug into the function.\r\n[reveal-answer q=\"792137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"792137\"]\r\n\r\n&nbsp;\r\n<table style=\"width: 385px;\" summary=\"The table shows 8 rows and 3 columns. The entries in the first row are: x; y = negative x plus 2; and (x, y). The entries in the second row are: negative 5; y = the opposite of negative 5 plus 2 = 7; (-5, 7). The entries in the third row are: negative 3; y = the opposite of negative 3 plus 2 = 5; (-3, 5). The entries in the fourth row are: -1; y = the opposite of negative 1 plus 2 = 3; (-1, 3). The entries in the fifth row are: 0; y = opposite of zero plus 2 = 2; (0, 2). The entries in the sixth row are: 1; y = the opposite of 1 plus 2 = 1; (1, 1). The entries in the seventh row are: 3; y = the opposite of 3 plus 2 = negative 1; (3, -1). The entries in the eighth row are: 5; y = the opposite of 5 plus 2 = negative 3; (5, -3).\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(x)=-x+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]-5[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(-5)=-\\left(-5\\right)+2=7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]-3[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(-3)=-\\left(-3\\right)+2=5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(-1)=-\\left(-1\\right)+2=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(0)=-\\left(0\\right)+2=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(1)=-\\left(1\\right)+2=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(3)=-\\left(3\\right)+2=-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 117.656px;\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 244.656px;\">[latex]f(5)=-\\left(5\\right)+2=-3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]219319[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]219320[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Use Sigma (Summation) Notation<\/h2>\r\n<strong>Summation notation <\/strong> (also known as summation notation) is used to make it easier to write lengthy sums. The Greek capital letter [latex]\\Sigma[\/latex], sigma, is used to express long sums of values in a compact form. For example, if we want to add all the integers from 1 to 20 without sigma notation, we have to write\r\n<div id=\"fs-id1170572139479\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572184284\">We could probably skip writing a couple of terms and write<\/p>\r\n\r\n<div id=\"fs-id1170572550692\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1+2+3+4+\\cdots+19+20[\/latex],<\/div>\r\nwhich is better, but still cumbersome. With sigma notation, we write this sum as\r\n<div id=\"fs-id1170572553991\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\sum_{i=1}^{20} i[\/latex],<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572132999\">which is much more compact.<\/p>\r\n<p id=\"fs-id1170571654708\">Typically, sigma notation is presented in the form<\/p>\r\n\r\n<div id=\"fs-id1170571602104\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\sum_{i=1}^{n} a_i[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572108202\">where [latex]a_i[\/latex] describes the terms to be added, and the [latex]i[\/latex] is called the <span class=\"no-emphasis\"><em>index<\/em><\/span>. Each term is evaluated, then we sum all the values, beginning with the value when [latex]i=1[\/latex] and ending with the value when [latex]i=n[\/latex]. For example, an expression like [latex]\\displaystyle\\sum_{i=2}^{7} s_i[\/latex] is interpreted as [latex]s_2+s_3+s_4+s_5+s_6+s_7[\/latex]. Note that the index is used only to keep track of the terms to be added; it does not factor into the calculation of the sum itself. The index is therefore called a <span class=\"no-emphasis\"><em>dummy variable<\/em><\/span>. We can use any letter we like for the index. Typically, mathematicians use [latex]i[\/latex], [latex]j[\/latex], [latex]k[\/latex], [latex]m[\/latex], and [latex]n[\/latex] for indices.<\/p>\r\n\r\n<div class=\"textbox\">\r\n<h3>A General Note: Summation Notation<\/h3>\r\nThe sum of the first [latex]n[\/latex] terms of a <strong>series <\/strong>can be expressed in <strong>summation notation<\/strong> as follows:\r\n<p style=\"text-align: center;\">[latex]\\sum _{i=1}^{n}{a}_{i}[\/latex]<\/p>\r\nThis notation tells us to find the sum of [latex]{a}_{i}[\/latex] from [latex]i=1[\/latex] to [latex]i=n[\/latex].\r\n\r\n[latex]k[\/latex] is called the <strong>index of summation<\/strong>, 1 is the <strong>lower limit of summation<\/strong>, and [latex]n[\/latex] is the <strong>upper limit of summation<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: EXpanding Summation Notation<\/h3>\r\nEvaluate [latex]\\sum _{i=3}^{7}{i}^{2}[\/latex].\r\n\r\n[reveal-answer q=\"14937\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"14937\"]\r\n\r\nAccording to the notation, the lower limit of summation is 3 and the upper limit is 7. So we need to find the sum of [latex]{i}^{2}[\/latex] from [latex]i=3[\/latex] to [latex]i=7[\/latex]. We find the terms of the series by substituting [latex]i=3\\text{,}4\\text{,}5\\text{,}6[\/latex], and [latex]7[\/latex] into the function [latex]{i}^{2}[\/latex]. We add the terms to find the sum.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sum _{i=3}^{7}{i}^{2}&amp; ={3}^{2}+{4}^{2}+{5}^{2}+{6}^{2}+{7}^{2}\\hfill \\\\ \\hfill &amp; =9+16+25+36+49\\hfill \\\\ \\hfill &amp; =135\\hfill \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nEvaluate [latex]\\sum _{i=2}^{5}\\left(3i - 1\\right)[\/latex].\r\n\r\n[reveal-answer q=\"812548\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"812548\"]\r\n\r\n38\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]222190[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Find the Area of a Trapezoid<\/h2>\r\nA trapezoid is a four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base [latex]b[\/latex] and the larger base [latex]B[\/latex]. The height, [latex]h[\/latex], of a trapezoid is the distance between the two bases as shown in the image below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223946\/CNX_BMath_Figure_09_04_052.png\" alt=\"A trapezoid is shown. The top is labeled b and marked as the smaller base. The bottom is labeled B and marked as the larger base. A vertical line forms a right angle with both bases and is marked as h.\" data-media-type=\"image\/png\" \/>\r\n\r\nThe formula for the area of a trapezoid is:\r\n\r\n[latex]Area_{trapezoid}=\\dfrac{1}{2}h(b+B)[\/latex]\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Area of a Trapezoid<\/h3>\r\nFind the area of a trapezoid whose height is 6 inches and whose bases are 14 and 11 inches.\r\n\r\n[reveal-answer q=\"149999\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"149999\"]\r\n<table id=\"eip-id1168468452905\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Step 1.\u00a0<strong>Read<\/strong>\u00a0the problem. Draw the figure and label it with the given information.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223952\/CNX_BMath_Figure_09_04_080_img-01.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2.\u00a0<strong>Identify<\/strong>\u00a0what you are looking for.<\/td>\r\n<td>the area of the trapezoid<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3.\u00a0<strong>Name.<\/strong>\u00a0Choose a variable to represent it.<\/td>\r\n<td>Let\u00a0<span id=\"MathJax-Element-13-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"font-family: proxima-nova, sans-serif; -webkit-font-smoothing: subpixel-antialiased; padding: 1px 0px; margin: 0px; font-size: 12.5568px; vertical-align: baseline; background: transparent; border: 0px; outline: 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;\/p&gt; &lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;A&lt;\/mi&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mtext&gt;the area&lt;\/mtext&gt;&lt;\/math&gt; &lt;p&gt;\"><span id=\"MJXc-Node-67\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-68\" class=\"mjx-mrow\"><span id=\"MJXc-Node-69\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">A<\/span><\/span><span id=\"MJXc-Node-70\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-71\" class=\"mjx-mtext MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">the area<\/span><\/span><\/span><\/span><\/span>\r\n\r\n<math><mi>A<\/mi><mo>=<\/mo><mtext>the area<\/mtext><\/math>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4.<strong>Translate.<\/strong>Write the appropriate formula.\r\n\r\nSubstitute.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223953\/CNX_BMath_Figure_09_04_080_img-02.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\">Step 5.\u00a0<strong>Solve<\/strong>\u00a0the equation.<\/td>\r\n<td><span id=\"MathJax-Element-14-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"font-family: proxima-nova, sans-serif; -webkit-font-smoothing: subpixel-antialiased; padding: 1px 0px; margin: 0px; font-size: 12.4416px; vertical-align: baseline; background: transparent; border: 0px; outline: 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;\/p&gt; &lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;A&lt;\/mi&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mfrac&gt;&lt;mo&gt;&amp;#x22C5;&lt;\/mo&gt;&lt;mn&gt;6&lt;\/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;\/mo&gt;&lt;mn&gt;25&lt;\/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;\/mo&gt;&lt;\/math&gt; &lt;p&gt;\"><span id=\"MJXc-Node-72\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-73\" class=\"mjx-mrow\"><span id=\"MJXc-Node-74\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">A<\/span><\/span><span id=\"MJXc-Node-75\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-76\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-77\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-78\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-79\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-80\" class=\"mjx-mn MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-81\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-82\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">25<\/span><\/span><span id=\"MJXc-Node-83\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><\/span><\/span>\r\n\r\n<math><mi>A<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>\u22c5<\/mo><mn>6<\/mn><mo>(<\/mo><mn>25<\/mn><mo>)<\/mo><\/math><span id=\"MathJax-Element-15-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"font-family: proxima-nova, sans-serif; -webkit-font-smoothing: subpixel-antialiased; padding: 1px 0px; margin: 0px; font-size: 12.5568px; vertical-align: baseline; background: transparent; border: 0px; outline: 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;\/p&gt; &lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;A&lt;\/mi&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;\/mo&gt;&lt;mn&gt;25&lt;\/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;\/mo&gt;&lt;\/math&gt; &lt;p&gt;\"><span id=\"MJXc-Node-84\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-85\" class=\"mjx-mrow\"><span id=\"MJXc-Node-86\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">A<\/span><\/span><span id=\"MJXc-Node-87\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-88\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-89\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-90\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">25<\/span><\/span><span id=\"MJXc-Node-91\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><\/span><\/span>\r\n\r\n<math><mi>A<\/mi><mo>=<\/mo><mn>3<\/mn><mo>(<\/mo><mn>25<\/mn><mo>)<\/mo><\/math>&nbsp;\r\n\r\n<span id=\"MathJax-Element-16-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"font-family: proxima-nova, sans-serif; -webkit-font-smoothing: subpixel-antialiased; padding: 1px 0px; margin: 0px; font-size: 12.5568px; vertical-align: baseline; background: transparent; border: 0px; outline: 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;\/p&gt; &lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;A&lt;\/mi&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mn&gt;75&lt;\/mn&gt;&lt;\/math&gt; &lt;p&gt;\"><span id=\"MJXc-Node-92\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-93\" class=\"mjx-mrow\"><span id=\"MJXc-Node-94\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">A<\/span><\/span><span id=\"MJXc-Node-95\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-96\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">75<\/span><\/span><\/span><\/span><\/span>\r\n\r\n<math><mi>A<\/mi><mo>=<\/mo><mn>75<\/mn><\/math>\u00a0square inches<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6.\u00a0<strong>Check:<\/strong>\u00a0Is this answer reasonable?<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]1109[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Complete a table values with solutions to an equation<\/li>\n<li>Use sigma (summation) notation to calculate sums and powers of integers<\/li>\n<li>Find the area of a trapezoid<\/li>\n<\/ul>\n<\/div>\n<p>In the Numerical Integration section, we will estimate the value of definite integrals using a variety of estimation rules. Here we will review some essential topics that will help us to better understand the various estimation rules.<\/p>\n<h2>Complete a Table of Function Values<\/h2>\n<div>\n<p>A table of values can be used to organize the\u00a0<em>y<\/em>-values or function values that result from plugging specific\u00a0<em>x<\/em>-values into a function&#8217;s equation.<\/p>\n<p>Suppose we want to determine the various values of the equation [latex]f(x)=2x - 1[\/latex] at certain values of <em>x<\/em>. We can begin by substituting a value for <em>x<\/em> into the equation and determining the resulting value of the function. The table below\u00a0lists some values of <em>x<\/em> from \u20133 to 3 and the resulting function values.<\/p>\n<table style=\"width: 411px; height: 84px;\" summary=\"This is a table with 8 rows and 3 columns. The first row has columns labeled: x, y = 2x-1, (x, y). The entries in the second row are: negative 3; y = 2 times negative 3 minus 1 = negative 7; (-3, -7). The entries in the third row are: negative 2; y = 2 times negative 2 minus 1 = negative 5; (-2, -5). The entries in the fourth row are: negative1; y = 2 times negative 1 minus 1 = negative 3; (-1, -3). The entries in the fifth row are: 0; y = 2 times 0 minus 1 = negative 1; (0, -1). The entries in the sixth row are: 1; y = 2 times 1 minus 1 = 1; (1, 1). The entries in the seventh row are: 2; y = 2 times 2 minus 1 = 3; (2, 3). The entries in the eight row are: 3, y = 2 times 3 minus 1 = 5; (3,5)\">\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"width: 116.5px; height: 12px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(x)=2x - 1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 116.5px; height: 12px;\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(-3)=2\\left(-3\\right)-1=-7[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 116.5px; height: 12px;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(-2)=2\\left(-2\\right)-1=-5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 116.5px; height: 12px;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(-1)=2\\left(-1\\right)-1=-3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 116.5px; height: 12px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(0)=2\\left(0\\right)-1=-1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 116.5px; height: 12px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(2)=2\\left(2\\right)-1=3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 116.5px; height: 12px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 269.5px; height: 12px;\">[latex]f(3)=2\\left(3\\right)-1=5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can look for trends among our function values by looking at the table. For example, in this case, as\u00a0<em>x<\/em>-values increase, so do the\u00a0<em>y<\/em>-values. Also, it seems reasonable to assume, based on the table, an\u00a0<em>x<\/em>-value of 1 would result in a function value of 1. You can verify this for yourself by plugging 1 into [latex]f(x)=2x - 1[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Completing a table of function values<\/h3>\n<p>Create a table of function values for [latex]f(x)=-x+2[\/latex]. Use various integers\u00a0from -5 to 5 as the\u00a0<em>x<\/em>-values you plug into the function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q792137\">Show Solution<\/span><\/p>\n<div id=\"q792137\" class=\"hidden-answer\" style=\"display: none\">\n<p>&nbsp;<\/p>\n<table style=\"width: 385px;\" summary=\"The table shows 8 rows and 3 columns. The entries in the first row are: x; y = negative x plus 2; and (x, y). The entries in the second row are: negative 5; y = the opposite of negative 5 plus 2 = 7; (-5, 7). The entries in the third row are: negative 3; y = the opposite of negative 3 plus 2 = 5; (-3, 5). The entries in the fourth row are: -1; y = the opposite of negative 1 plus 2 = 3; (-1, 3). The entries in the fifth row are: 0; y = opposite of zero plus 2 = 2; (0, 2). The entries in the sixth row are: 1; y = the opposite of 1 plus 2 = 1; (1, 1). The entries in the seventh row are: 3; y = the opposite of 3 plus 2 = negative 1; (3, -1). The entries in the eighth row are: 5; y = the opposite of 5 plus 2 = negative 3; (5, -3).\">\n<tbody>\n<tr>\n<td style=\"width: 117.656px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(x)=-x+2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 117.656px;\">[latex]-5[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(-5)=-\\left(-5\\right)+2=7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 117.656px;\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(-3)=-\\left(-3\\right)+2=5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 117.656px;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(-1)=-\\left(-1\\right)+2=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 117.656px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(0)=-\\left(0\\right)+2=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 117.656px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(1)=-\\left(1\\right)+2=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 117.656px;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(3)=-\\left(3\\right)+2=-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 117.656px;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 244.656px;\">[latex]f(5)=-\\left(5\\right)+2=-3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm219319\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=219319&theme=oea&iframe_resize_id=ohm219319&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm219320\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=219320&theme=oea&iframe_resize_id=ohm219320&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Use Sigma (Summation) Notation<\/h2>\n<p><strong>Summation notation <\/strong> (also known as summation notation) is used to make it easier to write lengthy sums. The Greek capital letter [latex]\\Sigma[\/latex], sigma, is used to express long sums of values in a compact form. For example, if we want to add all the integers from 1 to 20 without sigma notation, we have to write<\/p>\n<div id=\"fs-id1170572139479\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572184284\">We could probably skip writing a couple of terms and write<\/p>\n<div id=\"fs-id1170572550692\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]1+2+3+4+\\cdots+19+20[\/latex],<\/div>\n<p>which is better, but still cumbersome. With sigma notation, we write this sum as<\/p>\n<div id=\"fs-id1170572553991\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\sum_{i=1}^{20} i[\/latex],<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572132999\">which is much more compact.<\/p>\n<p id=\"fs-id1170571654708\">Typically, sigma notation is presented in the form<\/p>\n<div id=\"fs-id1170571602104\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\displaystyle\\sum_{i=1}^{n} a_i[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572108202\">where [latex]a_i[\/latex] describes the terms to be added, and the [latex]i[\/latex] is called the <span class=\"no-emphasis\"><em>index<\/em><\/span>. Each term is evaluated, then we sum all the values, beginning with the value when [latex]i=1[\/latex] and ending with the value when [latex]i=n[\/latex]. For example, an expression like [latex]\\displaystyle\\sum_{i=2}^{7} s_i[\/latex] is interpreted as [latex]s_2+s_3+s_4+s_5+s_6+s_7[\/latex]. Note that the index is used only to keep track of the terms to be added; it does not factor into the calculation of the sum itself. The index is therefore called a <span class=\"no-emphasis\"><em>dummy variable<\/em><\/span>. We can use any letter we like for the index. Typically, mathematicians use [latex]i[\/latex], [latex]j[\/latex], [latex]k[\/latex], [latex]m[\/latex], and [latex]n[\/latex] for indices.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Summation Notation<\/h3>\n<p>The sum of the first [latex]n[\/latex] terms of a <strong>series <\/strong>can be expressed in <strong>summation notation<\/strong> as follows:<\/p>\n<p style=\"text-align: center;\">[latex]\\sum _{i=1}^{n}{a}_{i}[\/latex]<\/p>\n<p>This notation tells us to find the sum of [latex]{a}_{i}[\/latex] from [latex]i=1[\/latex] to [latex]i=n[\/latex].<\/p>\n<p>[latex]k[\/latex] is called the <strong>index of summation<\/strong>, 1 is the <strong>lower limit of summation<\/strong>, and [latex]n[\/latex] is the <strong>upper limit of summation<\/strong>.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: EXpanding Summation Notation<\/h3>\n<p>Evaluate [latex]\\sum _{i=3}^{7}{i}^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q14937\">Show Solution<\/span><\/p>\n<div id=\"q14937\" class=\"hidden-answer\" style=\"display: none\">\n<p>According to the notation, the lower limit of summation is 3 and the upper limit is 7. So we need to find the sum of [latex]{i}^{2}[\/latex] from [latex]i=3[\/latex] to [latex]i=7[\/latex]. We find the terms of the series by substituting [latex]i=3\\text{,}4\\text{,}5\\text{,}6[\/latex], and [latex]7[\/latex] into the function [latex]{i}^{2}[\/latex]. We add the terms to find the sum.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\sum _{i=3}^{7}{i}^{2}& ={3}^{2}+{4}^{2}+{5}^{2}+{6}^{2}+{7}^{2}\\hfill \\\\ \\hfill & =9+16+25+36+49\\hfill \\\\ \\hfill & =135\\hfill \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Evaluate [latex]\\sum _{i=2}^{5}\\left(3i - 1\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q812548\">Show Solution<\/span><\/p>\n<div id=\"q812548\" class=\"hidden-answer\" style=\"display: none\">\n<p>38<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm222190\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=222190&theme=oea&iframe_resize_id=ohm222190\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Find the Area of a Trapezoid<\/h2>\n<p>A trapezoid is a four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base [latex]b[\/latex] and the larger base [latex]B[\/latex]. The height, [latex]h[\/latex], of a trapezoid is the distance between the two bases as shown in the image below.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223946\/CNX_BMath_Figure_09_04_052.png\" alt=\"A trapezoid is shown. The top is labeled b and marked as the smaller base. The bottom is labeled B and marked as the larger base. A vertical line forms a right angle with both bases and is marked as h.\" data-media-type=\"image\/png\" \/><\/p>\n<p>The formula for the area of a trapezoid is:<\/p>\n<p>[latex]Area_{trapezoid}=\\dfrac{1}{2}h(b+B)[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Area of a Trapezoid<\/h3>\n<p>Find the area of a trapezoid whose height is 6 inches and whose bases are 14 and 11 inches.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q149999\">Show Solution<\/span><\/p>\n<div id=\"q149999\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168468452905\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\" data-label=\"\">\n<tbody>\n<tr>\n<td>Step 1.\u00a0<strong>Read<\/strong>\u00a0the problem. Draw the figure and label it with the given information.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223952\/CNX_BMath_Figure_09_04_080_img-01.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2.\u00a0<strong>Identify<\/strong>\u00a0what you are looking for.<\/td>\n<td>the area of the trapezoid<\/td>\n<\/tr>\n<tr>\n<td>Step 3.\u00a0<strong>Name.<\/strong>\u00a0Choose a variable to represent it.<\/td>\n<td>Let\u00a0<span id=\"MathJax-Element-13-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"font-family: proxima-nova, sans-serif; -webkit-font-smoothing: subpixel-antialiased; padding: 1px 0px; margin: 0px; font-size: 12.5568px; vertical-align: baseline; background: transparent; border: 0px; outline: 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;\/p&gt; &lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;A&lt;\/mi&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mtext&gt;the area&lt;\/mtext&gt;&lt;\/math&gt; &lt;p&gt;\"><span id=\"MJXc-Node-67\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-68\" class=\"mjx-mrow\"><span id=\"MJXc-Node-69\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">A<\/span><\/span><span id=\"MJXc-Node-70\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-71\" class=\"mjx-mtext MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">the area<\/span><\/span><\/span><\/span><\/span><\/p>\n<p><math><mi>A<\/mi><mo>=<\/mo><mtext>the area<\/mtext><\/math>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 4.<strong>Translate.<\/strong>Write the appropriate formula.<\/p>\n<p>Substitute.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223953\/CNX_BMath_Figure_09_04_080_img-02.png\" alt=\".\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Step 5.\u00a0<strong>Solve<\/strong>\u00a0the equation.<\/td>\n<td><span id=\"MathJax-Element-14-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"font-family: proxima-nova, sans-serif; -webkit-font-smoothing: subpixel-antialiased; padding: 1px 0px; margin: 0px; font-size: 12.4416px; vertical-align: baseline; background: transparent; border: 0px; outline: 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;\/p&gt; &lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;A&lt;\/mi&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mfrac&gt;&lt;mo&gt;&amp;#x22C5;&lt;\/mo&gt;&lt;mn&gt;6&lt;\/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;\/mo&gt;&lt;mn&gt;25&lt;\/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;\/mo&gt;&lt;\/math&gt; &lt;p&gt;\"><span id=\"MJXc-Node-72\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-73\" class=\"mjx-mrow\"><span id=\"MJXc-Node-74\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">A<\/span><\/span><span id=\"MJXc-Node-75\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-76\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-77\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-78\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-79\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u22c5<\/span><\/span><span id=\"MJXc-Node-80\" class=\"mjx-mn MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><span id=\"MJXc-Node-81\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-82\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">25<\/span><\/span><span id=\"MJXc-Node-83\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<p><math><mi>A<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo>\u22c5<\/mo><mn>6<\/mn><mo>(<\/mo><mn>25<\/mn><mo>)<\/mo><\/math><span id=\"MathJax-Element-15-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"font-family: proxima-nova, sans-serif; -webkit-font-smoothing: subpixel-antialiased; padding: 1px 0px; margin: 0px; font-size: 12.5568px; vertical-align: baseline; background: transparent; border: 0px; outline: 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;\/p&gt; &lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;A&lt;\/mi&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;\/mo&gt;&lt;mn&gt;25&lt;\/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;\/mo&gt;&lt;\/math&gt; &lt;p&gt;\"><span id=\"MJXc-Node-84\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-85\" class=\"mjx-mrow\"><span id=\"MJXc-Node-86\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">A<\/span><\/span><span id=\"MJXc-Node-87\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-88\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-89\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-90\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">25<\/span><\/span><span id=\"MJXc-Node-91\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><\/span><\/span><\/p>\n<p><math><mi>A<\/mi><mo>=<\/mo><mn>3<\/mn><mo>(<\/mo><mn>25<\/mn><mo>)<\/mo><\/math>&nbsp;<\/p>\n<p><span id=\"MathJax-Element-16-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"font-family: proxima-nova, sans-serif; -webkit-font-smoothing: subpixel-antialiased; padding: 1px 0px; margin: 0px; font-size: 12.5568px; vertical-align: baseline; background: transparent; border: 0px; outline: 0px; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;\/p&gt; &lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;A&lt;\/mi&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mn&gt;75&lt;\/mn&gt;&lt;\/math&gt; &lt;p&gt;\"><span id=\"MJXc-Node-92\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-93\" class=\"mjx-mrow\"><span id=\"MJXc-Node-94\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">A<\/span><\/span><span id=\"MJXc-Node-95\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-96\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">75<\/span><\/span><\/span><\/span><\/span><\/p>\n<p><math><mi>A<\/mi><mo>=<\/mo><mn>75<\/mn><\/math>\u00a0square inches<\/td>\n<\/tr>\n<tr>\n<td>Step 6.\u00a0<strong>Check:<\/strong>\u00a0Is this answer reasonable?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm1109\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1109&theme=oea&iframe_resize_id=ohm1109\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1988\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Modification and Revision. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/.\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/.<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":349141,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Modification and Revision\",\"author\":\"\",\"organization\":\"Lumen 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