{"id":673,"date":"2021-05-10T18:54:20","date_gmt":"2021-05-10T18:54:20","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=673"},"modified":"2021-11-17T02:34:12","modified_gmt":"2021-11-17T02:34:12","slug":"summary-of-improper-integrals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-improper-integrals\/","title":{"raw":"Summary of Improper Integrals","rendered":"Summary of Improper Integrals"},"content":{"raw":"<section id=\"fs-id1165042657932\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\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<\/div>\r\n<\/section><section id=\"fs-id1165042390895\" class=\"key-equations\" data-depth=\"1\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1165042390902\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Improper integrals<\/strong><span data-type=\"newline\">\r\n<\/span>\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-id1165042505590\" class=\"section-exercises\" data-depth=\"1\"><\/section>\r\n<div data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\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<\/div>","rendered":"<section id=\"fs-id1165042657932\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\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<\/div>\n<\/section>\n<section id=\"fs-id1165042390895\" class=\"key-equations\" data-depth=\"1\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1165042390902\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Improper integrals<\/strong><span data-type=\"newline\"><br \/>\n<\/span><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-id1165042505590\" class=\"section-exercises\" data-depth=\"1\"><\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\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<\/div>\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-673\">\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":416434,"menu_order":27,"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-673","chapter","type-chapter","status-publish","hentry"],"part":158,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/673","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\/416434"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/673\/revisions"}],"predecessor-version":[{"id":834,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/673\/revisions\/834"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/158"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/673\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=673"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=673"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=673"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=673"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}