{"id":155166,"date":"2022-03-26T00:20:40","date_gmt":"2022-03-25T18:50:40","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/riemann-integral\/"},"modified":"2023-01-02T13:07:58","modified_gmt":"2023-01-02T07:37:58","slug":"riemann-integral","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/riemann-integral\/","title":{"rendered":"Riemann Integral"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/riemann-integral\/#Explain_in_Detail_Important_Riemann_Sum_Terms\" title=\"Explain in Detail :Important Riemann Sum Terms\">Explain in Detail :Important Riemann Sum Terms<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/riemann-integral\/#Riemann_Sum\" title=\"Riemann Sum\">Riemann Sum<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/riemann-integral\/#Riemann_Integral_Formula\" title=\"Riemann Integral Formula\">Riemann Integral Formula<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/riemann-integral\/#Properties_of_Riemann_Integral\" title=\"Properties of Riemann Integral\">Properties of Riemann Integral<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/riemann-integral\/#Riemann_Sum_Example\" title=\"Riemann Sum Example\">Riemann Sum Example<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Explain_in_Detail_Important_Riemann_Sum_Terms\"><\/span>Explain in Detail :Important Riemann Sum Terms<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A Riemann sum is a way to approximate the area under a curve by adding up the areas of many small squares. The approximation gets better as the squares get smaller.<\/p>\n<p>There are three important terms used in Riemann sums:<\/p>\n<p>&#8211; The function being approximated is called the &#8220;integrand.&#8221;<\/p>\n<p>&#8211; The area of a square is called the &#8220;height&#8221; or &#8220;dx.&#8221;<\/p>\n<p>&#8211; The number of squares used to approximate the area is called the &#8220;numerator.&#8221;<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-155165 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/riemann-integral.jpg\" alt=\"\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/riemann-integral.jpg?v=1648234234 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/riemann-integral-300x212.jpg?v=1648234234 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Riemann_Sum\"><\/span>Riemann Sum<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A Riemann sum is a method for approximating the area under a curve. The curve is divided into a number of sections, and the area of each section is calculated. The total area is then approximated by adding up the individual areas.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Riemann_Integral_Formula\"><\/span>Riemann Integral Formula<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Riemann integral formula states that the integral of a function can be found by breaking the function into a sequence of pieces, each of which is approximated by a Riemann sum.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Properties_of_Riemann_Integral\"><\/span>Properties of Riemann Integral<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>1. The Riemann integral is a function from a subset of the real numbers to the real numbers.<\/p>\n<p>2. The Riemann integral is defined as the limit of a sequence of Riemann sums.<\/p>\n<p>3. The Riemann integral is a linear operator.<\/p>\n<p>4. The Riemann integral is a positive operator.<\/p>\n<p>5. The Riemann integral is a monotone operator.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Riemann_Sum_Example\"><\/span>Riemann Sum Example<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The following table gives the area of a rectangle under a curve for six different points along the curve.<\/p>\n<p>\\begin{array}{ccc}<\/p>\n<p>x &amp; y &amp; A \\\\<\/p>\n<p>1 &amp; 1.5 &amp; 3.75 \\\\<\/p>\n<p>2 &amp; 2.5 &amp; 5.625 \\\\<\/p>\n<p>3 &amp; 3.5 &amp; 7.5 \\\\<\/p>\n<p>4 &amp; 4.5 &amp; 9.375 \\\\<\/p>\n<p>5 &amp; 5.5 &amp; 11.25 \\\\<\/p>\n<p>6 &amp; 6.5 &amp; 13.125<\/p>\n<p>\\end{array}<\/p>\n<p>We can approximate the area under the curve by using a Riemann sum. In this example, we will use the leftmost point, the rightmost point, and four points in between.<\/p>\n<p>\\begin{array}{ccc}<\/p>\n<p>x &amp; y &amp; A \\\\<\/p>\n<p>1 &amp; 1.5 &amp; 3.75 \\\\<\/p>\n<p>2 &amp; 2.5 &amp; 5.625 \\\\<\/p>\n<p>3 &amp; 3.5 &amp; 7.5 \\\\<\/p>\n<p>4 &amp; 4.5 &amp; 9.375 \\\\<\/p>\n<p>5 &amp; 5.5 &amp; 11.25 \\\\<\/p>\n<p>6 &amp; 6.5 &amp; 13.125<\/p>\n<p>\\end{array}<\/p>\n<p>The Riemann sum for this example is:<\/p>\n<p>\\begin{align}<\/p>\n<p>A &amp;= 3.75+5.625+7.5+9.375+11.25+13<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Explain in Detail :Important Riemann Sum Terms A Riemann sum is a way to approximate the area under a curve [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Riemann Integral","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learn about Riemann Integral topic of Maths in details explained by subject experts on infinitylearn.com. 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