{"id":741,"date":"2025-06-20T17:09:16","date_gmt":"2025-06-20T17:09:16","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=741"},"modified":"2025-07-22T14:04:00","modified_gmt":"2025-07-22T14:04:00","slug":"numerical-and-improper-integration-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/numerical-and-improper-integration-cheat-sheet\/","title":{"raw":"Numerical and Improper Integration: Cheat Sheet","rendered":"Numerical and Improper Integration: Cheat Sheet"},"content":{"raw":"<strong>Numerical Integration Methods<\/strong>\r\n<ul id=\"fs-id1165041757005\" data-bullet-style=\"bullet\">\r\n \t<li>We can use numerical integration to estimate the values of definite integrals when a closed form of the integral is difficult to find or when an approximate value only of the definite integral is needed.<\/li>\r\n \t<li>The most commonly used techniques for numerical integration are the midpoint rule, trapezoidal rule, and Simpson\u2019s rule.<\/li>\r\n \t<li>The midpoint rule approximates the definite integral using rectangular regions whereas the trapezoidal rule approximates the definite integral using trapezoidal approximations.<\/li>\r\n \t<li>Simpson\u2019s rule approximates the definite integral by first approximating the original function using piecewise quadratic functions.<\/li>\r\n<\/ul>\r\n<strong>Improper Integrals<\/strong>\r\n<ul id=\"fs-id1165042390874\" data-bullet-style=\"bullet\">\r\n \t<li>Integrals of functions over infinite intervals are defined in terms of limits.<\/li>\r\n \t<li>Integrals of functions over an interval for which the function has a discontinuity at an endpoint may be defined in terms of limits.<\/li>\r\n \t<li>The convergence or divergence of an improper integral may be determined by comparing it with the value of an improper integral for which the convergence or divergence is known.<\/li>\r\n<\/ul>\r\n<section id=\"fs-id1165041757034\" class=\"key-equations\" data-depth=\"1\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1165041830110\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Midpoint rule<\/strong><span data-type=\"newline\">\r\n<\/span> [latex]{M}_{n}=\\displaystyle\\sum _{i=1}^{n}f\\left({m}_{i}\\right)\\Delta x[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Trapezoidal rule<\/strong><span data-type=\"newline\">\r\n<\/span> [latex]{T}_{n}=\\frac{1}{2}\\Delta x\\left(f\\left({x}_{0}\\right)+2f\\left({x}_{1}\\right)+2f\\left({x}_{2}\\right)+\\cdots +2f\\left({x}_{n - 1}\\right)+f\\left({x}_{n}\\right)\\right)[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Simpson\u2019s rule<\/strong><span data-type=\"newline\">\r\n<\/span> [latex]{S}_{n}=\\frac{\\Delta x}{3}\\left(f\\left({x}_{0}\\right)+4f\\left({x}_{1}\\right)+2f\\left({x}_{2}\\right)+4f\\left({x}_{3}\\right)+2f\\left({x}_{4}\\right)+4f\\left({x}_{5}\\right)+\\cdots +2f\\left({x}_{n - 2}\\right)+4f\\left({x}_{n - 1}\\right)+f\\left({x}_{n}\\right)\\right)[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Error bound for midpoint rule<\/strong><span data-type=\"newline\">\r\n<\/span> [latex]\\text{Error in }{M}_{n}\\le \\frac{M{\\left(b-a\\right)}^{3}}{24{n}^{2}}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Error bound for trapezoidal rule<\/strong><span data-type=\"newline\">\r\n<\/span> [latex]\\text{Error in }{T}_{n}\\le \\frac{M{\\left(b-a\\right)}^{3}}{12{n}^{2}}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Error bound for Simpson\u2019s rule<\/strong><span data-type=\"newline\">\r\n<\/span> [latex]\\text{Error in }{S}_{n}\\le \\frac{M{\\left(b-a\\right)}^{5}}{180{n}^{4}}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Improper integrals<\/strong>\r\n[latex]\\begin{array}{c}{\\displaystyle\\int }_{a}^{+\\infty }f\\left(x\\right)dx=\\underset{t\\to \\text{+}\\infty }{\\text{lim}}{\\displaystyle\\int }_{a}^{t}f\\left(x\\right)dx\\hfill \\\\ {\\displaystyle\\int }_{\\text{-}\\infty }^{b}f\\left(x\\right)dx=\\underset{t\\to \\text{-}\\infty }{\\text{lim}}{\\displaystyle\\int }_{t}^{b}f\\left(x\\right)dx\\hfill \\\\ {\\displaystyle\\int }_{\\text{-}\\infty }^{+\\infty }f\\left(x\\right)dx={\\displaystyle\\int }_{\\text{-}\\infty }^{0}f\\left(x\\right)dx+{\\displaystyle\\int }_{0}^{+\\infty }f\\left(x\\right)dx\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section id=\"fs-id1165040744472\" class=\"section-exercises\" data-depth=\"1\"><\/section>\r\n<div data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165040778942\">\r\n \t<dt>absolute error<\/dt>\r\n \t<dd id=\"fs-id1165040778947\">if [latex]B[\/latex] is an estimate of some quantity having an actual value of [latex]A[\/latex], then the absolute error is given by [latex]|A-B|[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165039564320\">\r\n \t<dt>improper integral<\/dt>\r\n \t<dd id=\"fs-id1165039564326\">an integral over an infinite interval or an integral of a function containing an infinite discontinuity on the interval; an improper integral is defined in terms of a limit. The improper integral converges if this limit is a finite real number; otherwise, the improper integral diverges<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042040454\">\r\n \t<dt>midpoint rule<\/dt>\r\n \t<dd id=\"fs-id1165042040459\">a rule that uses a Riemann sum of the form [latex]{M}_{n}=\\displaystyle\\sum _{i=1}^{n}f\\left({m}_{i}\\right)\\Delta x[\/latex], where [latex]{m}_{i}[\/latex] is the midpoint of the <em data-effect=\"italics\">i<\/em>th subinterval to approximate [latex]{\\displaystyle\\int }_{a}^{b}f\\left(x\\right)dx[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165040796205\">\r\n \t<dt>numerical integration<\/dt>\r\n \t<dd id=\"fs-id1165040720628\">the variety of numerical methods used to estimate the value of a definite integral, including the midpoint rule, trapezoidal rule, and Simpson\u2019s rule<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165040720635\">\r\n \t<dt>relative error<\/dt>\r\n \t<dd id=\"fs-id1165040720640\">error as a percentage of the absolute value, given by [latex]|\\frac{A-B}{A}|=|\\frac{A-B}{A}|\\cdot 100\\text{%}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165040720689\">\r\n \t<dt>Simpson\u2019s rule<\/dt>\r\n \t<dd id=\"fs-id1165041825148\">a rule that approximates [latex]{\\displaystyle\\int }_{a}^{b}f\\left(x\\right)dx[\/latex] using the integrals of a piecewise quadratic function. The approximation [latex]{S}_{n}[\/latex] to [latex]{\\displaystyle\\int }_{a}^{b}f\\left(x\\right)dx[\/latex] is given by [latex]{S}_{n}=\\frac{\\Delta x}{3}\\left(\\begin{array}{c}f\\left({x}_{0}\\right)+4f\\left({x}_{1}\\right)+2f\\left({x}_{2}\\right)+4f\\left({x}_{3}\\right)+2f\\left({x}_{4}\\right)+4f\\left({x}_{5}\\right)\\\\ +\\cdots +2f\\left({x}_{n - 2}\\right)+4f\\left({x}_{n - 1}\\right)+f\\left({x}_{n}\\right)\\end{array}\\right)[\/latex] trapezoidal rule a rule that approximates [latex]{\\displaystyle\\int }_{a}^{b}f\\left(x\\right)dx[\/latex] using trapezoids<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<p><strong>Numerical Integration Methods<\/strong><\/p>\n<ul id=\"fs-id1165041757005\" data-bullet-style=\"bullet\">\n<li>We can use numerical integration to estimate the values of definite integrals when a closed form of the integral is difficult to find or when an approximate value only of the definite integral is needed.<\/li>\n<li>The most commonly used techniques for numerical integration are the midpoint rule, trapezoidal rule, and Simpson\u2019s rule.<\/li>\n<li>The midpoint rule approximates the definite integral using rectangular regions whereas the trapezoidal rule approximates the definite integral using trapezoidal approximations.<\/li>\n<li>Simpson\u2019s rule approximates the definite integral by first approximating the original function using piecewise quadratic functions.<\/li>\n<\/ul>\n<p><strong>Improper Integrals<\/strong><\/p>\n<ul id=\"fs-id1165042390874\" data-bullet-style=\"bullet\">\n<li>Integrals of functions over infinite intervals are defined in terms of limits.<\/li>\n<li>Integrals of functions over an interval for which the function has a discontinuity at an endpoint may be defined in terms of limits.<\/li>\n<li>The convergence or divergence of an improper integral may be determined by comparing it with the value of an improper integral for which the convergence or divergence is known.<\/li>\n<\/ul>\n<section id=\"fs-id1165041757034\" class=\"key-equations\" data-depth=\"1\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1165041830110\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Midpoint rule<\/strong><span data-type=\"newline\"><br \/>\n<\/span> [latex]{M}_{n}=\\displaystyle\\sum _{i=1}^{n}f\\left({m}_{i}\\right)\\Delta x[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Trapezoidal rule<\/strong><span data-type=\"newline\"><br \/>\n<\/span> [latex]{T}_{n}=\\frac{1}{2}\\Delta x\\left(f\\left({x}_{0}\\right)+2f\\left({x}_{1}\\right)+2f\\left({x}_{2}\\right)+\\cdots +2f\\left({x}_{n - 1}\\right)+f\\left({x}_{n}\\right)\\right)[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Simpson\u2019s rule<\/strong><span data-type=\"newline\"><br \/>\n<\/span> [latex]{S}_{n}=\\frac{\\Delta x}{3}\\left(f\\left({x}_{0}\\right)+4f\\left({x}_{1}\\right)+2f\\left({x}_{2}\\right)+4f\\left({x}_{3}\\right)+2f\\left({x}_{4}\\right)+4f\\left({x}_{5}\\right)+\\cdots +2f\\left({x}_{n - 2}\\right)+4f\\left({x}_{n - 1}\\right)+f\\left({x}_{n}\\right)\\right)[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Error bound for midpoint rule<\/strong><span data-type=\"newline\"><br \/>\n<\/span> [latex]\\text{Error in }{M}_{n}\\le \\frac{M{\\left(b-a\\right)}^{3}}{24{n}^{2}}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Error bound for trapezoidal rule<\/strong><span data-type=\"newline\"><br \/>\n<\/span> [latex]\\text{Error in }{T}_{n}\\le \\frac{M{\\left(b-a\\right)}^{3}}{12{n}^{2}}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Error bound for Simpson\u2019s rule<\/strong><span data-type=\"newline\"><br \/>\n<\/span> [latex]\\text{Error in }{S}_{n}\\le \\frac{M{\\left(b-a\\right)}^{5}}{180{n}^{4}}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Improper integrals<\/strong><br \/>\n[latex]\\begin{array}{c}{\\displaystyle\\int }_{a}^{+\\infty }f\\left(x\\right)dx=\\underset{t\\to \\text{+}\\infty }{\\text{lim}}{\\displaystyle\\int }_{a}^{t}f\\left(x\\right)dx\\hfill \\\\ {\\displaystyle\\int }_{\\text{-}\\infty }^{b}f\\left(x\\right)dx=\\underset{t\\to \\text{-}\\infty }{\\text{lim}}{\\displaystyle\\int }_{t}^{b}f\\left(x\\right)dx\\hfill \\\\ {\\displaystyle\\int }_{\\text{-}\\infty }^{+\\infty }f\\left(x\\right)dx={\\displaystyle\\int }_{\\text{-}\\infty }^{0}f\\left(x\\right)dx+{\\displaystyle\\int }_{0}^{+\\infty }f\\left(x\\right)dx\\hfill \\end{array}[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id1165040744472\" class=\"section-exercises\" data-depth=\"1\"><\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165040778942\">\n<dt>absolute error<\/dt>\n<dd id=\"fs-id1165040778947\">if [latex]B[\/latex] is an estimate of some quantity having an actual value of [latex]A[\/latex], then the absolute error is given by [latex]|A-B|[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165039564320\">\n<dt>improper integral<\/dt>\n<dd id=\"fs-id1165039564326\">an integral over an infinite interval or an integral of a function containing an infinite discontinuity on the interval; an improper integral is defined in terms of a limit. The improper integral converges if this limit is a finite real number; otherwise, the improper integral diverges<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042040454\">\n<dt>midpoint rule<\/dt>\n<dd id=\"fs-id1165042040459\">a rule that uses a Riemann sum of the form [latex]{M}_{n}=\\displaystyle\\sum _{i=1}^{n}f\\left({m}_{i}\\right)\\Delta x[\/latex], where [latex]{m}_{i}[\/latex] is the midpoint of the <em data-effect=\"italics\">i<\/em>th subinterval to approximate [latex]{\\displaystyle\\int }_{a}^{b}f\\left(x\\right)dx[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165040796205\">\n<dt>numerical integration<\/dt>\n<dd id=\"fs-id1165040720628\">the variety of numerical methods used to estimate the value of a definite integral, including the midpoint rule, trapezoidal rule, and Simpson\u2019s rule<\/dd>\n<\/dl>\n<dl id=\"fs-id1165040720635\">\n<dt>relative error<\/dt>\n<dd id=\"fs-id1165040720640\">error as a percentage of the absolute value, given by [latex]|\\frac{A-B}{A}|=|\\frac{A-B}{A}|\\cdot 100\\text{%}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165040720689\">\n<dt>Simpson\u2019s rule<\/dt>\n<dd id=\"fs-id1165041825148\">a rule that approximates [latex]{\\displaystyle\\int }_{a}^{b}f\\left(x\\right)dx[\/latex] using the integrals of a piecewise quadratic function. The approximation [latex]{S}_{n}[\/latex] to [latex]{\\displaystyle\\int }_{a}^{b}f\\left(x\\right)dx[\/latex] is given by [latex]{S}_{n}=\\frac{\\Delta x}{3}\\left(\\begin{array}{c}f\\left({x}_{0}\\right)+4f\\left({x}_{1}\\right)+2f\\left({x}_{2}\\right)+4f\\left({x}_{3}\\right)+2f\\left({x}_{4}\\right)+4f\\left({x}_{5}\\right)\\\\ +\\cdots +2f\\left({x}_{n - 2}\\right)+4f\\left({x}_{n - 1}\\right)+f\\left({x}_{n}\\right)\\end{array}\\right)[\/latex] trapezoidal rule a rule that approximates [latex]{\\displaystyle\\int }_{a}^{b}f\\left(x\\right)dx[\/latex] using trapezoids<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":15,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":667,"module-header":"- Select Header -","content_attributions":[],"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\/741"}],"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\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/741\/revisions"}],"predecessor-version":[{"id":1360,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/741\/revisions\/1360"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/667"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/741\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=741"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=741"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=741"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=741"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}