{"id":3968,"date":"2021-05-25T17:39:14","date_gmt":"2021-05-25T17:39:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-approximating-areas\/"},"modified":"2021-07-03T19:12:22","modified_gmt":"2021-07-03T19:12:22","slug":"review-for-approximating-areas","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-approximating-areas\/","title":{"raw":"Skills Review for Approximating Areas","rendered":"Skills Review for Approximating Areas"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Calculate the area of a rectangle<\/li>\r\n \t<li>Use summation notation<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Approximating Areas section, we will estimate the area under a curve by dividing the region under the curve into rectangles. Here we will review how to find the area of a rectangle and how to expand summation notation.\r\n<h2>Find the Area of a Rectangle<\/h2>\r\nThe <b>area of a rectangle<\/b>\u00a0can be found using the following formula:\r\n<p style=\"text-align: center;\">[latex]A=lw[\/latex]<\/p>\r\nwhere [latex]l[\/latex] is the length and [latex]w[\/latex] is the width of the rectangle.\r\n\r\nNote: Sometimes the area of a rectangle can be found using the formula [latex]A=bh[\/latex] where [latex]b[\/latex] is the base and [latex]h[\/latex] is the height of the rectangle.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding The Area of a Rectangle<\/h3>\r\nFind the area of a rectangle with a length of 3 inches and a width of 2 inches.\r\n\r\n[reveal-answer q=\"133742\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133742\"]\r\n\r\nWe will use the formula for calculating the area of a rectangle. In this case, [latex]l=3[\/latex] and [latex]w=2[\/latex].\r\n\r\nSo, we have:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A=lw\\\\A=3(2)\\\\A=6 \\text{ inches}^2\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding The Area of a Rectangle<\/h3>\r\nFind the area of a rectangle with a base of 2 centimeters and a height of 7 centimeters.\r\n\r\n[reveal-answer q=\"133743\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133743\"]\r\n\r\nWe will use the formula for calculating the area of a rectangle. In this case, [latex]b=2[\/latex] and [latex]h=7[\/latex].\r\n\r\nSo, we have:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A=bh\\\\A=2(7)\\\\A=14 \\text{ centimeters}^2\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the area of a rectangle with a base of 7 centimeters and a height of 9 inches.\r\n\r\n[reveal-answer q=\"133744\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133744\"]\r\n<p style=\"text-align: left;\">[latex]A=63 \\text{ centimeters}^2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Expand Sigma (Summation) Notation<\/h2>\r\n<strong>Summation notation <\/strong>is used to represent long sums of values in a compact form. Summation notation is often known as sigma notation because it uses the Greek capital letter <strong>sigma<\/strong>\u00a0to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms of the sum. An explicit formula for each term of the series is given to the right of the sigma. A variable called the <strong>index of summation <\/strong>is written below the sigma. The index of summation is set equal to the <strong>lower limit of summation<\/strong>, which is the number used to generate the first term of the sum. The number above the sigma, called the <strong>upper limit of summation<\/strong>, is the number used to generate the last term of the sum.\r\n\r\nIf we interpret the given notation, we see that it asks us to find the sum of the terms in the series [latex]{a}_{i}=2i[\/latex] for [latex]i=1[\/latex] through [latex]i=5[\/latex]. We can begin by substituting the terms for [latex]i[\/latex] and listing out the terms.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l} {a}_{1}=2\\left(1\\right)=2 \\\\ {a}_{2}=2\\left(2\\right)=4\\hfill \\\\ {a}_{3}=2\\left(3\\right)=6\\hfill \\\\ {a}_{4}=2\\left(4\\right)=8\\hfill \\\\ {a}_{5}=2\\left(5\\right)=10\\hfill \\end{array}[\/latex]<\/div>\r\nWe can find the sum by adding the terms:\r\n<div style=\"text-align: center;\">[latex]\\displaystyle\\sum _{i=1}^{5}2i=2+4+6+8+10=30[\/latex]<\/div>\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Calculate the area of a rectangle<\/li>\n<li>Use summation notation<\/li>\n<\/ul>\n<\/div>\n<p>In the Approximating Areas section, we will estimate the area under a curve by dividing the region under the curve into rectangles. Here we will review how to find the area of a rectangle and how to expand summation notation.<\/p>\n<h2>Find the Area of a Rectangle<\/h2>\n<p>The <b>area of a rectangle<\/b>\u00a0can be found using the following formula:<\/p>\n<p style=\"text-align: center;\">[latex]A=lw[\/latex]<\/p>\n<p>where [latex]l[\/latex] is the length and [latex]w[\/latex] is the width of the rectangle.<\/p>\n<p>Note: Sometimes the area of a rectangle can be found using the formula [latex]A=bh[\/latex] where [latex]b[\/latex] is the base and [latex]h[\/latex] is the height of the rectangle.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding The Area of a Rectangle<\/h3>\n<p>Find the area of a rectangle with a length of 3 inches and a width of 2 inches.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133742\">Show Solution<\/span><\/p>\n<div id=\"q133742\" class=\"hidden-answer\" style=\"display: none\">\n<p>We will use the formula for calculating the area of a rectangle. In this case, [latex]l=3[\/latex] and [latex]w=2[\/latex].<\/p>\n<p>So, we have:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A=lw\\\\A=3(2)\\\\A=6 \\text{ inches}^2\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding The Area of a Rectangle<\/h3>\n<p>Find the area of a rectangle with a base of 2 centimeters and a height of 7 centimeters.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133743\">Show Solution<\/span><\/p>\n<div id=\"q133743\" class=\"hidden-answer\" style=\"display: none\">\n<p>We will use the formula for calculating the area of a rectangle. In this case, [latex]b=2[\/latex] and [latex]h=7[\/latex].<\/p>\n<p>So, we have:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}A=bh\\\\A=2(7)\\\\A=14 \\text{ centimeters}^2\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the area of a rectangle with a base of 7 centimeters and a height of 9 inches.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133744\">Show Solution<\/span><\/p>\n<div id=\"q133744\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">[latex]A=63 \\text{ centimeters}^2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Expand Sigma (Summation) Notation<\/h2>\n<p><strong>Summation notation <\/strong>is used to represent long sums of values in a compact form. Summation notation is often known as sigma notation because it uses the Greek capital letter <strong>sigma<\/strong>\u00a0to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms of the sum. An explicit formula for each term of the series is given to the right of the sigma. A variable called the <strong>index of summation <\/strong>is written below the sigma. The index of summation is set equal to the <strong>lower limit of summation<\/strong>, which is the number used to generate the first term of the sum. The number above the sigma, called the <strong>upper limit of summation<\/strong>, is the number used to generate the last term of the sum.<\/p>\n<p>If we interpret the given notation, we see that it asks us to find the sum of the terms in the series [latex]{a}_{i}=2i[\/latex] for [latex]i=1[\/latex] through [latex]i=5[\/latex]. We can begin by substituting the terms for [latex]i[\/latex] and listing out the terms.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l} {a}_{1}=2\\left(1\\right)=2 \\\\ {a}_{2}=2\\left(2\\right)=4\\hfill \\\\ {a}_{3}=2\\left(3\\right)=6\\hfill \\\\ {a}_{4}=2\\left(4\\right)=8\\hfill \\\\ {a}_{5}=2\\left(5\\right)=10\\hfill \\end{array}[\/latex]<\/div>\n<p>We can find the sum by adding the terms:<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle\\sum _{i=1}^{5}2i=2+4+6+8+10=30[\/latex]<\/div>\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]\\displaystyle\\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]\\displaystyle\\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]\\displaystyle\\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\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-3968\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><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":17533,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Modification and Revision \",\"author\":\"\",\"organization\":\"Lumen 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