{"id":1119,"date":"2021-06-30T17:01:55","date_gmt":"2021-06-30T17:01:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-approximating-areas\/"},"modified":"2021-11-17T01:34:40","modified_gmt":"2021-11-17T01:34:40","slug":"summary-of-approximating-areas","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-approximating-areas\/","title":{"raw":"Summary of Approximating Areas","rendered":"Summary of Approximating Areas"},"content":{"raw":"<div id=\"fs-id1170571769572\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170571769579\">\r\n \t<li>The use of sigma (summation) notation of the form [latex]\\displaystyle\\sum_{i=1}^{n}a_i[\/latex] is useful for expressing long sums of values in compact form.<\/li>\r\n \t<li>For a continuous function defined over an interval [latex][a,b][\/latex], the process of dividing the interval into [latex]n[\/latex] equal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region.<\/li>\r\n \t<li>The width of each rectangle is [latex]\\Delta x=\\dfrac{b-a}{n}[\/latex]<\/li>\r\n \t<li>Riemann sums are expressions of the form [latex]\\displaystyle\\sum_{i=1}^{n} f(x_i^*)\\Delta x[\/latex], and can be used to estimate the area under the curve [latex]y=f(x)[\/latex]. Left- and right-endpoint approximations are special kinds of Riemann sums where the values of [latex]\\{x_i^*\\}[\/latex] are chosen to be the left or right endpoints of the subintervals, respectively.<\/li>\r\n \t<li>Riemann sums allow for much flexibility in choosing the set of points [latex]\\{x_i^*\\}[\/latex] at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1170572565393\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170572565400\">\r\n \t<li><strong>Properties of Sigma Notation<\/strong>\r\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}c=nc[\/latex]\r\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}ca_i=c\\underset{i=1}{\\overset{n}{\\Sigma}}a_i[\/latex]\r\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}(a_i+b_i)=\\underset{i=1}{\\overset{n}{\\Sigma}}a_i+\\underset{i=1}{\\overset{n}{\\Sigma}}b_i[\/latex]\r\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}(a_i-b_i)=\\underset{i=1}{\\overset{n}{\\Sigma}}a_i-\\underset{i=1}{\\overset{n}{\\Sigma}}b_i[\/latex]\r\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}a_i=\\underset{i=1}{\\overset{m}{\\Sigma}}a_i+\\underset{i=m+1}{\\overset{n}{\\Sigma}}a_i[\/latex]<\/li>\r\n \t<li><strong>Sums and Powers of Integers<\/strong>\r\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i=1+2+\\cdots+n=\\frac{n(n+1)}{2}[\/latex]\r\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i^2=1^2+2^2+\\cdots+n^2=\\frac{n(n+1)(2n+1)}{6}[\/latex]\r\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i^3=1^3+2^3+\\cdots+n^3=\\frac{n^2(n+1)^2}{4}[\/latex]<\/li>\r\n \t<li><strong>Left-Endpoint Approximation<\/strong>\r\n[latex]A \\approx L_n=f(x_0)\\Delta x+f(x_1)\\Delta x+\\cdots+f(x_{n-1})\\Delta x=\\underset{i=1}{\\overset{n}{\\Sigma}}f(x_{i-1})\\Delta x[\/latex]<\/li>\r\n \t<li><strong>Right-Endpoint Approximation<\/strong>\r\n[latex]A \\approx R_n=f(x_1)\\Delta x+f(x_2)\\Delta x+\\cdots+f(x_n)\\Delta x=\\underset{i=1}{\\overset{n}{\\Sigma}}f(x_i)\\Delta x[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170571577527\" class=\"definition\">\r\n \t<dt>left-endpoint approximation<\/dt>\r\n \t<dd id=\"fs-id1170571577533\">an approximation of the area under a curve computed by using the left endpoint of each subinterval to calculate the height of the vertical sides of each rectangle<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571577538\" class=\"definition\">\r\n \t<dt>lower sum<\/dt>\r\n \t<dd id=\"fs-id1170571577544\">a sum obtained by using the minimum value of [latex]f(x)[\/latex] on each subinterval<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571577561\" class=\"definition\">\r\n \t<dt>partition<\/dt>\r\n \t<dd id=\"fs-id1170571577566\">a set of points that divides an interval into subintervals<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571577570\" class=\"definition\">\r\n \t<dt>regular partition<\/dt>\r\n \t<dd id=\"fs-id1170571577576\">a partition in which the subintervals all have the same width<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571577580\" class=\"definition\">\r\n \t<dt>riemann sum<\/dt>\r\n \t<dd id=\"fs-id1170571577585\">an estimate of the area under the curve of the form [latex]A\\approx \\underset{i=1}{\\overset{n}{\\Sigma}}f(x_i^*)\\Delta x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572373515\" class=\"definition\">\r\n \t<dt>right-endpoint approximation<\/dt>\r\n \t<dd id=\"fs-id1170572373521\">the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of each rectangle<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572373527\" class=\"definition\">\r\n \t<dt>sigma notation<\/dt>\r\n \t<dd id=\"fs-id1170572373532\">(also, <strong>summation notation<\/strong>) the Greek letter sigma ([latex]\\Sigma[\/latex]) indicates addition of the values; the values of the index above and below the sigma indicate where to begin the summation and where to end it<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572373545\" class=\"definition\">\r\n \t<dt>upper sum<\/dt>\r\n \t<dd id=\"fs-id1170572373550\">a sum obtained by using the maximum value of [latex]f(x)[\/latex] on each subinterval<\/dd>\r\n<\/dl>","rendered":"<div id=\"fs-id1170571769572\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170571769579\">\n<li>The use of sigma (summation) notation of the form [latex]\\displaystyle\\sum_{i=1}^{n}a_i[\/latex] is useful for expressing long sums of values in compact form.<\/li>\n<li>For a continuous function defined over an interval [latex][a,b][\/latex], the process of dividing the interval into [latex]n[\/latex] equal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region.<\/li>\n<li>The width of each rectangle is [latex]\\Delta x=\\dfrac{b-a}{n}[\/latex]<\/li>\n<li>Riemann sums are expressions of the form [latex]\\displaystyle\\sum_{i=1}^{n} f(x_i^*)\\Delta x[\/latex], and can be used to estimate the area under the curve [latex]y=f(x)[\/latex]. Left- and right-endpoint approximations are special kinds of Riemann sums where the values of [latex]\\{x_i^*\\}[\/latex] are chosen to be the left or right endpoints of the subintervals, respectively.<\/li>\n<li>Riemann sums allow for much flexibility in choosing the set of points [latex]\\{x_i^*\\}[\/latex] at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1170572565393\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572565400\">\n<li><strong>Properties of Sigma Notation<\/strong><br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}c=nc[\/latex]<br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}ca_i=c\\underset{i=1}{\\overset{n}{\\Sigma}}a_i[\/latex]<br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}(a_i+b_i)=\\underset{i=1}{\\overset{n}{\\Sigma}}a_i+\\underset{i=1}{\\overset{n}{\\Sigma}}b_i[\/latex]<br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}(a_i-b_i)=\\underset{i=1}{\\overset{n}{\\Sigma}}a_i-\\underset{i=1}{\\overset{n}{\\Sigma}}b_i[\/latex]<br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}a_i=\\underset{i=1}{\\overset{m}{\\Sigma}}a_i+\\underset{i=m+1}{\\overset{n}{\\Sigma}}a_i[\/latex]<\/li>\n<li><strong>Sums and Powers of Integers<\/strong><br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i=1+2+\\cdots+n=\\frac{n(n+1)}{2}[\/latex]<br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i^2=1^2+2^2+\\cdots+n^2=\\frac{n(n+1)(2n+1)}{6}[\/latex]<br \/>\n[latex]\\underset{i=1}{\\overset{n}{\\Sigma}}i^3=1^3+2^3+\\cdots+n^3=\\frac{n^2(n+1)^2}{4}[\/latex]<\/li>\n<li><strong>Left-Endpoint Approximation<\/strong><br \/>\n[latex]A \\approx L_n=f(x_0)\\Delta x+f(x_1)\\Delta x+\\cdots+f(x_{n-1})\\Delta x=\\underset{i=1}{\\overset{n}{\\Sigma}}f(x_{i-1})\\Delta x[\/latex]<\/li>\n<li><strong>Right-Endpoint Approximation<\/strong><br \/>\n[latex]A \\approx R_n=f(x_1)\\Delta x+f(x_2)\\Delta x+\\cdots+f(x_n)\\Delta x=\\underset{i=1}{\\overset{n}{\\Sigma}}f(x_i)\\Delta x[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170571577527\" class=\"definition\">\n<dt>left-endpoint approximation<\/dt>\n<dd id=\"fs-id1170571577533\">an approximation of the area under a curve computed by using the left endpoint of each subinterval to calculate the height of the vertical sides of each rectangle<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571577538\" class=\"definition\">\n<dt>lower sum<\/dt>\n<dd id=\"fs-id1170571577544\">a sum obtained by using the minimum value of [latex]f(x)[\/latex] on each subinterval<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571577561\" class=\"definition\">\n<dt>partition<\/dt>\n<dd id=\"fs-id1170571577566\">a set of points that divides an interval into subintervals<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571577570\" class=\"definition\">\n<dt>regular partition<\/dt>\n<dd id=\"fs-id1170571577576\">a partition in which the subintervals all have the same width<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571577580\" class=\"definition\">\n<dt>riemann sum<\/dt>\n<dd id=\"fs-id1170571577585\">an estimate of the area under the curve of the form [latex]A\\approx \\underset{i=1}{\\overset{n}{\\Sigma}}f(x_i^*)\\Delta x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572373515\" class=\"definition\">\n<dt>right-endpoint approximation<\/dt>\n<dd id=\"fs-id1170572373521\">the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of each rectangle<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572373527\" class=\"definition\">\n<dt>sigma notation<\/dt>\n<dd id=\"fs-id1170572373532\">(also, <strong>summation notation<\/strong>) the Greek letter sigma ([latex]\\Sigma[\/latex]) indicates addition of the values; the values of the index above and below the sigma indicate where to begin the summation and where to end it<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572373545\" class=\"definition\">\n<dt>upper sum<\/dt>\n<dd id=\"fs-id1170572373550\">a sum obtained by using the maximum value of [latex]f(x)[\/latex] on each subinterval<\/dd>\n<\/dl>\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-1119\">\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>Calculus Volume 2. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/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":6,"template":"","meta":{"_candela_citation":"{\"1\":{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\"}}","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1119","chapter","type-chapter","status-publish","hentry"],"part":1113,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1119","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1119\/revisions"}],"predecessor-version":[{"id":2463,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1119\/revisions\/2463"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/1113"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1119\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1119"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1119"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1119"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1119"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}