{"id":192,"date":"2019-08-27T20:02:23","date_gmt":"2019-08-27T20:02:23","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=192"},"modified":"2020-06-18T16:22:45","modified_gmt":"2020-06-18T16:22:45","slug":"riemann-sums","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/riemann-sums\/","title":{"rendered":"Riemann Sums"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Riemann Sums &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n<!------------------------>\n\nSuppose that a function $f$ is continuous and non-negative on an\ninterval $[a,b]$.\n\n<\/p>\n\n\n\n<p>\n<b>Let&#8217;s compute the area of the region $R$ bounded above by the curve\n$y=f(x)$, below by the x-axis, and on the sides by the lines $x=a$ and\n$x=b$.<\/b>\n\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"508\" height=\"218\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/sums_figure1.gif?resize=508%2C218&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-393\"\/><\/figure><\/div>\n\n\n\n<p>\nWe will obtain this area as the limit of a sum of areas of rectangles\nas follows:\n\n<\/p>\n\n\n\n<p>\nFirst, we will divide the interval $[a,b]$ into $n$ subintervals\n\\[ [x_0, x_1], [x_1, x_2], \\ldots, [x_{n-1}, x_n] \\]\nwhere $a = x_0 &lt; x_1 &lt; \\ldots &lt; x_n = b$.  (This is called a\n<b>partition<\/b> of the interval.)\nThe intervals need not all be the same length, so call the lengths of\nthe intervals $\\Delta x_1$, $\\Delta x_2$, \\ldots, $\\Delta x_n$,\nrespectively.  This partition divides the region $R$ into $n$ strips.\n\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"507\" height=\"217\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/sums_figure2.gif?resize=507%2C217&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-394\"\/><\/figure><\/div>\n\n\n\n<p>\nNext, let&#8217;s approximate each strip by a rectangle with height equal to\nthe height of the curve $y=f(x)$ at some arbitrary point in the\nsubinterval.  That is, for the first subinterval $[x_0, x_1]$, select\nsome $x_1^\\ast$ contained in that subinterval and use $f(x_1^\\ast)$ as\nthe height of the first rectangle.  The area of that rectangle is then\n$f(x_1^\\ast) \\Delta x_1$.\n\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"526\" height=\"219\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/sums_figure3.gif?resize=526%2C219&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-395\"\/><\/figure><\/div>\n\n\n\n<p>\nSimilarly, for each subinterval $[x_{i-1}, x_i]$, we will choose some\n$x_i^\\ast$ and calculate the area of the corresponding rectangle to be\n$f(x_i^\\ast) \\Delta x_i$.  The approximate area of the region $R$ is\nthen the sum $\\sum_{i=1}^n f(x_i^\\ast) \\Delta x_i$ of these\nrectangles.\n\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"507\" height=\"234\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/sums_figure4.gif?resize=507%2C234&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-396\"\/><\/figure><\/div>\n\n\n\n<p>\nDepending on what points we select for the $x_i^\\ast$, our estimate\nmay be too large or too small.  For example, if we choose each\n$x_i^\\ast$ to be the point in its subinterval giving the\n<i>maximum<\/i> height, we will <i>overestimate<\/i> the area of $R$.\n(This is called a <b>upper sum<\/b>.)  \n\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"257\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/sums_figure5.gif?resize=500%2C257&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-397\"\/><\/figure><\/div>\n\n\n\n<p>\nIf, on the other hand, we choose each $x_i^\\ast$ to be the point in its \nsubinterval giving the <i>mimimum<\/i> height, we will <i>underestimate<\/i>\nthe area of $R$. (This is called a <b>lower sum<\/b>.)  \n\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"235\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/sums_figure6.gif?resize=500%2C235&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-391\"\/><\/figure><\/div>\n\n\n\n<p>\nWhen the points $x_i^\\ast$ are chosen randomly, the sum\n$\\sum_{i=1}^n f(x_i^\\ast) \\Delta x_i$ is called a <b>Riemann Sum<\/b>  \n\n<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"507\" height=\"234\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus\/wp-content\/uploads\/sites\/3\/2019\/08\/sums_figure7.gif?resize=507%2C234&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-392\"\/><\/figure><\/div>\n\n\n\n<p>\nand will give an approximation for the area of $R$\nthat is in between the lower and upper sums.  The upper and lower sums\nmay be considered specific Riemann sums.\n\n<\/p>\n\n\n\n<p>\nAs we decrease the widths of the rectangles, we expect to be able to\napproximate the area of $R$ better.  In fact, as max\n$\\Delta x_i \\rightarrow 0$, we get the exact area of $R$, which we\ndenote by the definite integral $\\int_a^b f(x)\\,dx$.  That is,\n\\[\\int_a^b f(x)\\,dx =\n \\lim_{max \\Delta x_i\\rightarrow 0} \\left(\\sum_{i=1}^n f(x_i^\\ast)\\Delta x_i\\right).\\] \n\n<\/p>\n\n\n\n<center>\n<h4>Notes<\/h4>\n<\/center>\n\n\n\n<ul class=\"wp-block-list\"><li> This definition of the definite integral still holds if $f(x)$\n\t\tassumes both positive and negative values on $[a, b]$.  It even holds\n\t\tif $f(x)$ has finitely many discontinuities but is bounded.\n\t<br><br>\n\t<\/li><li> For a more rigorous treatment of Riemann sums, consult your\n\t\tcalculus text.\n<\/li><\/ul>\n\n\n\n<p>\nThe following Exploration allows you to approximate the area under various \ncurves under the interval $[0, 5]$. You can create a partition of the interval \nand view an upper sum, a lower sum, or another Riemann sum using that \npartition. The Exploration will give you the exact area and calculate the area \nof your approximation. To create a partition, choose which type of sum you \nwould like to see and click the mouse between the partition labels $x_0$ and \n$x_1$.\n\n<\/p>\n\n\n\n<p>\nLet $f$ be defined on $[a, b]$ and let ${x_0, x_1, \\ldots, x_n}$ be a\npartition of $[a, b]$.\n\n<\/p>\n\n\n\n<p>\nFor each $[x_{i-1}, x_i]$, let $x_i^\\ast \\in [x_{i-1}, x_i]$.\n\n<\/p>\n\n\n\n<p>\nThen the definite integral of $f$ over $[a, b]$ as defined as\n\\[\\int_a^b f(x)\\,dx =\n \\lim_{max \\Delta x_i\\rightarrow 0} \\left(\\sum_{i=1}^n f(x_i^\\ast)\\Delta x_i\\right).\\] \n\n<!------------------------>\n\n\n<br>\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ1710\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Riemann Sums &#8211; HMC Calculus Tutorial Suppose that a function $f$ is continuous and non-negative on an interval $[a,b]$. Let&#8217;s compute the area of the region $R$ bounded above by the curve $y=f(x)$, below by the x-axis, and on the sides by the lines $x=a$ and $x=b$. We will obtain this area as the limit&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":57,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-192","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/192","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=192"}],"version-history":[{"count":7,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/192\/revisions"}],"predecessor-version":[{"id":1219,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/192\/revisions\/1219"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/57"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=192"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=192"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}