{"id":376,"date":"2025-02-13T19:44:26","date_gmt":"2025-02-13T19:44:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/approximating-areas-learn-it-1\/"},"modified":"2025-02-13T19:44:26","modified_gmt":"2025-02-13T19:44:26","slug":"approximating-areas-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/approximating-areas-learn-it-1\/","title":{"raw":"Approximating Areas: Learn It 1","rendered":"Approximating Areas: Learn It 1"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Estimate the area under a curve by adding up the areas of rectangles<\/li>\n\t<li>Estimate the area under a curve using Riemann sums<\/li>\n<\/ul>\n<\/section>\n<h2>Approximating Area<\/h2>\n<p id=\"fs-id1170571599828\">Archimedes used a process known as the method of exhaustion to calculate the area of various shapes. This involved using smaller and smaller shapes to approximate the area more accurately. We can use a similar method to approximate the area under a curve, [latex]f(x)[\/latex], between [latex]a[\/latex] and b. By dividing the area into smaller rectangles, we get closer approximations.<\/p>\n<p>Let [latex]f(x)[\/latex] be a continuous, nonnegative function defined on the closed interval [latex][a,b][\/latex]. Our goal is to approximate the area [latex]A[\/latex] bounded by [latex]f(x)[\/latex] above, the [latex]x[\/latex]-axis below, the line [latex]x=a[\/latex] on the left, and the line [latex]x=b[\/latex] on the right (Figure 1).<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"448\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203841\/CNX_Calc_Figure_05_01_017.jpg\" alt=\"A graph in quadrant one of an area bounded by a generic curve f(x) at the top, the x-axis at the bottom, the line x = a to the left, and the line x = b to the right. About midway through, the concavity switches from concave down to concave up, and the function starts to increases shortly before the line x = b.\" width=\"448\" height=\"422\"> Figure 1. An area (shaded region) bounded by the curve [latex]f(x)[\/latex] at top, the x-axis at bottom, the line [latex]x=a[\/latex] to the left, and the line [latex]x=b[\/latex] at right.[\/caption]\n\n<p id=\"fs-id1170572419054\">To approximate the area under this curve, we use a geometric approach. By dividing the region into many small shapes with known area formulas, we can sum these areas to estimate the true area reasonably well.<\/p>\n<p>We begin by dividing the interval [latex][a,b][\/latex] into [latex]n[\/latex] subintervals of equal width, [latex]\\frac{b-a}{n}[\/latex]. We select equally spaced points [latex]x_0,x_1,x_2,\\cdots,x_n[\/latex] with [latex]x_0=a, \\, x_n=b[\/latex], and<\/p>\n<div id=\"fs-id1170571613009\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]x_i-x_{i-1}=\\dfrac{b-a}{n}[\/latex]<\/div>\n<p>for [latex]i=1,2,3,\\cdots,n[\/latex].<\/p>\n<p id=\"fs-id1170572621706\">The width of each subinterval is denoted as [latex]\\Delta x[\/latex], so [latex]\\Delta x=\\dfrac{b-a}{n}[\/latex] and the points are defined by<\/p>\n<div id=\"fs-id1170572223464\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]x_i=x_0+i \\Delta x[\/latex]<\/div>\n<p id=\"fs-id1170571696623\">for [latex]i=1,2,3,\\cdots,n[\/latex].<\/p>\n<p>This method of dividing an interval [latex][a,b][\/latex] into subintervals using equally spaced points is commonly used to approximate the area under a curve. Let's define some relevant terminology to make this process clearer.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div class=\"title\">\n<h3 style=\"text-align: left;\">partition<\/h3>\n\nA set of points [latex]P=\\{x_i\\}[\/latex] for [latex]i=0,1,2,\\cdots,n[\/latex] with [latex]a=x_0 &lt; x_1 &lt; x_2 &lt; \\cdots &lt; x_n=b[\/latex], which divides the interval [latex][a,b][\/latex] into subintervals of the form [latex][x_0,x_1], \\, [x_1,x_2],\\cdots,[x_{n-1},x_n][\/latex] is called a<strong> partition<\/strong> of [latex][a,b][\/latex]. If the subintervals all have the same width, the set of points forms a<strong> regular partition<\/strong> of the interval [latex][a,b][\/latex].<\/div>\n<\/section>\n<p id=\"fs-id1170572452131\">We can use this regular partition as the basis of a method for estimating the area under the curve. We next examine two methods: the <strong>left-endpoint approximation<\/strong> and the <strong>right-endpoint approximation<\/strong>.<\/p>\n<h3>Left-Endpoint Approximation<\/h3>\n<p id=\"fs-id1170571696639\">On each subinterval [latex][x_{i-1},x_i][\/latex] (for [latex]i=1,2,3,\\cdots,n[\/latex]), construct a rectangle with width [latex]\\Delta x[\/latex] and height equal to [latex]f(x_{i-1})[\/latex], which is the function value at the left endpoint of the subinterval. Then the area of this rectangle is [latex]f(x_{i-1})\\Delta x[\/latex].<\/p>\n<p>Adding the areas of all these rectangles, we get an approximate value for [latex]A[\/latex]. We use the notation [latex]L_n[\/latex] to denote that this is a left-endpoint approximation of [latex]A[\/latex] using [latex]n[\/latex] subintervals.<\/p>\n<div id=\"fs-id1170571645657\" class=\"equation\" style=\"text-align: center;\">[latex]\\begin{array}{ll} A \\approx L_n &amp; =f(x_0)\\Delta x+f(x_1)\\Delta x+\\cdots+f(x_{n-1})\\Delta x \\\\ &amp; =\\displaystyle\\sum_{i=1}^{n} f(x_{i-1})\\Delta x \\end{array}[\/latex]<\/div>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">left-endpoint approximation<\/h3>\n<div id=\"fs-id1170571645657\" class=\"equation\" style=\"text-align: left;\">In the left-endpoint approximation, we estimate the area under a curve by constructing rectangles whose heights are determined by the function values at the left endpoints of subintervals.<\/div>\n<div>&nbsp;<\/div>\n<div class=\"equation\" style=\"text-align: left;\">The approximation of the area [latex]A[\/latex] using [latex]n[\/latex] subintervals is given by the formula:<\/div>\n<div style=\"text-align: center;\"><br>\n[latex] A \\approx L_n = \\displaystyle\\sum_{i=1}^{n} f(x_{i-1})\\Delta x [\/latex]<\/div>\n<p>where [latex]\\Delta x =\\frac{b-a}{n}[\/latex] is the width of each subinterval, and [latex]x_{i-1}[\/latex] are the left endpoints of the subintervals.<\/p>\n<\/section>\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203844\/CNX_Calc_Figure_05_01_001.jpg\" alt=\"A diagram showing the left-endpoint approximation of area under a curve. Under a parabola with vertex on the y axis and above the x axis, rectangles are drawn between a=x0 on the origin and b = xn. The rectangles have endpoints at a=x0, x1, x2\u2026x(n-1), and b = xn, spaced equally. The height of each rectangle is determined by the value of the given function at the left endpoint of the rectangle.\" width=\"487\" height=\"241\"> Figure 2. In the left-endpoint approximation of area under a curve, the height of each rectangle is determined by the function value at the left of each subinterval.[\/caption]\n\n<section class=\"textbox questionHelp\">\n<p><strong>How to: Perform Left-Endpoint Approximation<\/strong><\/p>\n<ol>\n\t<li><strong>Identify the Interval and Function: <\/strong>Determine the interval [latex][a,b][\/latex] and the function [latex]f(x) [\/latex] you are working with.<\/li>\n\t<li><strong>Divide the Interval<\/strong>: Divide [latex][a,b][\/latex] into [latex]n[\/latex] subintervals of equal width [latex]\\Delta x =\\frac{b-a}{n}[\/latex].<\/li>\n\t<li><strong>Determine Left Endpoints:<\/strong> Identify the left endpoints of each subinterval: [latex]x_0,x_1,x_2,\\cdots,x_{n-1}[\/latex]<\/li>\n\t<li><strong>Evaluate the Function: <\/strong>Calculate the function values at each left endpoint: [latex]f(x_0), f(x_1), f(x_2), \\cdots, f(x_{n-1})[\/latex].<\/li>\n\t<li><strong>Calculate Rectangle Areas: <\/strong>Multiply each function value by the width [latex]\u0394x[\/latex] to get the area of each rectangle.<\/li>\n\t<li><strong>Sum the Areas<\/strong>: Add up all the rectangle areas to get the approximate area under the curve<center>[latex] A \\approx \\displaystyle\\sum_{i=1}^{n} f(x_{i-1})\\Delta x [\/latex]<\/center><\/li>\n<\/ol>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Estimate the area under a curve by adding up the areas of rectangles<\/li>\n<li>Estimate the area under a curve using Riemann sums<\/li>\n<\/ul>\n<\/section>\n<h2>Approximating Area<\/h2>\n<p id=\"fs-id1170571599828\">Archimedes used a process known as the method of exhaustion to calculate the area of various shapes. This involved using smaller and smaller shapes to approximate the area more accurately. We can use a similar method to approximate the area under a curve, [latex]f(x)[\/latex], between [latex]a[\/latex] and b. By dividing the area into smaller rectangles, we get closer approximations.<\/p>\n<p>Let [latex]f(x)[\/latex] be a continuous, nonnegative function defined on the closed interval [latex][a,b][\/latex]. Our goal is to approximate the area [latex]A[\/latex] bounded by [latex]f(x)[\/latex] above, the [latex]x[\/latex]-axis below, the line [latex]x=a[\/latex] on the left, and the line [latex]x=b[\/latex] on the right (Figure 1).<\/p>\n<figure style=\"width: 448px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203841\/CNX_Calc_Figure_05_01_017.jpg\" alt=\"A graph in quadrant one of an area bounded by a generic curve f(x) at the top, the x-axis at the bottom, the line x = a to the left, and the line x = b to the right. About midway through, the concavity switches from concave down to concave up, and the function starts to increases shortly before the line x = b.\" width=\"448\" height=\"422\" \/><figcaption class=\"wp-caption-text\">Figure 1. An area (shaded region) bounded by the curve [latex]f(x)[\/latex] at top, the x-axis at bottom, the line [latex]x=a[\/latex] to the left, and the line [latex]x=b[\/latex] at right.<\/figcaption><\/figure>\n<p id=\"fs-id1170572419054\">To approximate the area under this curve, we use a geometric approach. By dividing the region into many small shapes with known area formulas, we can sum these areas to estimate the true area reasonably well.<\/p>\n<p>We begin by dividing the interval [latex][a,b][\/latex] into [latex]n[\/latex] subintervals of equal width, [latex]\\frac{b-a}{n}[\/latex]. We select equally spaced points [latex]x_0,x_1,x_2,\\cdots,x_n[\/latex] with [latex]x_0=a, \\, x_n=b[\/latex], and<\/p>\n<div id=\"fs-id1170571613009\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]x_i-x_{i-1}=\\dfrac{b-a}{n}[\/latex]<\/div>\n<p>for [latex]i=1,2,3,\\cdots,n[\/latex].<\/p>\n<p id=\"fs-id1170572621706\">The width of each subinterval is denoted as [latex]\\Delta x[\/latex], so [latex]\\Delta x=\\dfrac{b-a}{n}[\/latex] and the points are defined by<\/p>\n<div id=\"fs-id1170572223464\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]x_i=x_0+i \\Delta x[\/latex]<\/div>\n<p id=\"fs-id1170571696623\">for [latex]i=1,2,3,\\cdots,n[\/latex].<\/p>\n<p>This method of dividing an interval [latex][a,b][\/latex] into subintervals using equally spaced points is commonly used to approximate the area under a curve. Let&#8217;s define some relevant terminology to make this process clearer.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div class=\"title\">\n<h3 style=\"text-align: left;\">partition<\/h3>\n<p>A set of points [latex]P=\\{x_i\\}[\/latex] for [latex]i=0,1,2,\\cdots,n[\/latex] with [latex]a=x_0 < x_1 < x_2 < \\cdots < x_n=b[\/latex], which divides the interval [latex][a,b][\/latex] into subintervals of the form [latex][x_0,x_1], \\, [x_1,x_2],\\cdots,[x_{n-1},x_n][\/latex] is called a<strong> partition<\/strong> of [latex][a,b][\/latex]. If the subintervals all have the same width, the set of points forms a<strong> regular partition<\/strong> of the interval [latex][a,b][\/latex].<\/div>\n<\/section>\n<p id=\"fs-id1170572452131\">We can use this regular partition as the basis of a method for estimating the area under the curve. We next examine two methods: the <strong>left-endpoint approximation<\/strong> and the <strong>right-endpoint approximation<\/strong>.<\/p>\n<h3>Left-Endpoint Approximation<\/h3>\n<p id=\"fs-id1170571696639\">On each subinterval [latex][x_{i-1},x_i][\/latex] (for [latex]i=1,2,3,\\cdots,n[\/latex]), construct a rectangle with width [latex]\\Delta x[\/latex] and height equal to [latex]f(x_{i-1})[\/latex], which is the function value at the left endpoint of the subinterval. Then the area of this rectangle is [latex]f(x_{i-1})\\Delta x[\/latex].<\/p>\n<p>Adding the areas of all these rectangles, we get an approximate value for [latex]A[\/latex]. We use the notation [latex]L_n[\/latex] to denote that this is a left-endpoint approximation of [latex]A[\/latex] using [latex]n[\/latex] subintervals.<\/p>\n<div id=\"fs-id1170571645657\" class=\"equation\" style=\"text-align: center;\">[latex]\\begin{array}{ll} A \\approx L_n & =f(x_0)\\Delta x+f(x_1)\\Delta x+\\cdots+f(x_{n-1})\\Delta x \\\\ & =\\displaystyle\\sum_{i=1}^{n} f(x_{i-1})\\Delta x \\end{array}[\/latex]<\/div>\n<section class=\"textbox keyTakeaway\">\n<h3 style=\"text-align: left;\">left-endpoint approximation<\/h3>\n<div class=\"equation\" style=\"text-align: left;\">In the left-endpoint approximation, we estimate the area under a curve by constructing rectangles whose heights are determined by the function values at the left endpoints of subintervals.<\/div>\n<div>&nbsp;<\/div>\n<div class=\"equation\" style=\"text-align: left;\">The approximation of the area [latex]A[\/latex] using [latex]n[\/latex] subintervals is given by the formula:<\/div>\n<div style=\"text-align: center;\">\n[latex]A \\approx L_n = \\displaystyle\\sum_{i=1}^{n} f(x_{i-1})\\Delta x[\/latex]<\/div>\n<p>where [latex]\\Delta x =\\frac{b-a}{n}[\/latex] is the width of each subinterval, and [latex]x_{i-1}[\/latex] are the left endpoints of the subintervals.<\/p>\n<\/section>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11203844\/CNX_Calc_Figure_05_01_001.jpg\" alt=\"A diagram showing the left-endpoint approximation of area under a curve. Under a parabola with vertex on the y axis and above the x axis, rectangles are drawn between a=x0 on the origin and b = xn. The rectangles have endpoints at a=x0, x1, x2\u2026x(n-1), and b = xn, spaced equally. The height of each rectangle is determined by the value of the given function at the left endpoint of the rectangle.\" width=\"487\" height=\"241\" \/><figcaption class=\"wp-caption-text\">Figure 2. In the left-endpoint approximation of area under a curve, the height of each rectangle is determined by the function value at the left of each subinterval.<\/figcaption><\/figure>\n<section class=\"textbox questionHelp\">\n<p><strong>How to: Perform Left-Endpoint Approximation<\/strong><\/p>\n<ol>\n<li><strong>Identify the Interval and Function: <\/strong>Determine the interval [latex][a,b][\/latex] and the function [latex]f(x)[\/latex] you are working with.<\/li>\n<li><strong>Divide the Interval<\/strong>: Divide [latex][a,b][\/latex] into [latex]n[\/latex] subintervals of equal width [latex]\\Delta x =\\frac{b-a}{n}[\/latex].<\/li>\n<li><strong>Determine Left Endpoints:<\/strong> Identify the left endpoints of each subinterval: [latex]x_0,x_1,x_2,\\cdots,x_{n-1}[\/latex]<\/li>\n<li><strong>Evaluate the Function: <\/strong>Calculate the function values at each left endpoint: [latex]f(x_0), f(x_1), f(x_2), \\cdots, f(x_{n-1})[\/latex].<\/li>\n<li><strong>Calculate Rectangle Areas: <\/strong>Multiply each function value by the width [latex]\u0394x[\/latex] to get the area of each rectangle.<\/li>\n<li><strong>Sum the Areas<\/strong>: Add up all the rectangle areas to get the approximate area under the curve\n<div style=\"text-align: center;\">[latex]A \\approx \\displaystyle\\sum_{i=1}^{n} f(x_{i-1})\\Delta x[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/section>\n","protected":false},"author":6,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"5.1 Approximating Areas\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":371,"module-header":"","content_attributions":[{"type":"cc","description":"Calculus Volume 1","author":"Gilbert Strang, Edwin (Jed) Herman","organization":"OpenStax","url":"https:\/\/openstax.org\/details\/books\/calculus-volume-1","project":"","license":"cc-by-nc-sa","license_terms":"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction"},{"type":"original","description":"5.1 Approximating Areas","author":"Ryan Melton","organization":"","url":"","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/376"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/376\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/371"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/376\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=376"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=376"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=376"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=376"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}