{"id":1487,"date":"2021-03-18T22:28:52","date_gmt":"2021-03-18T22:28:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1487"},"modified":"2021-03-18T22:28:52","modified_gmt":"2021-03-18T22:28:52","slug":"summary-of-physical-applications","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-physical-applications\/","title":{"raw":"Summary of Physical Applications","rendered":"Summary of Physical Applications"},"content":{"raw":"<div id=\"fs-id1167791543265\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1167791543272\">\r\n \t<li>Several physical applications of the definite integral are common in engineering and physics.<\/li>\r\n \t<li>Definite integrals can be used to determine the mass of an object if its density function is known.<\/li>\r\n \t<li>Work can also be calculated from integrating a force function, or when counteracting the force of gravity, as in a pumping problem.<\/li>\r\n \t<li>Definite integrals can also be used to calculate the force exerted on an object submerged in a liquid.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1167791543292\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1167791543298\">\r\n \t<li><strong>Mass of a one-dimensional object<\/strong>\r\n[latex]m={\\displaystyle\\int }_{a}^{b}\\rho (x)dx[\/latex]<\/li>\r\n \t<li><strong>Mass of a circular object<\/strong>\r\n[latex]m={\\displaystyle\\int }_{0}^{r}2\\pi x\\rho (x)dx[\/latex]<\/li>\r\n \t<li><strong>Work done on an object<\/strong>\r\n[latex]W={\\displaystyle\\int }_{a}^{b}F(x)dx[\/latex]<\/li>\r\n \t<li><strong>Hydrostatic force on a plate<\/strong>\r\n[latex]F={\\displaystyle\\int }_{a}^{b}\\rho w(x)s(x)dx[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1167793976005\" class=\"definition\">\r\n \t<dt>density function<\/dt>\r\n \t<dd id=\"fs-id1167793609788\">a density function describes how mass is distributed throughout an object; it can be a linear density, expressed in terms of mass per unit length; an area density, expressed in terms of mass per unit area; or a volume density, expressed in terms of mass per unit volume; weight-density is also used to describe weight (rather than mass) per unit volume<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167793609795\" class=\"definition\">\r\n \t<dt>Hooke\u2019s law<\/dt>\r\n \t<dd id=\"fs-id1167793609800\">this law states that the force required to compress (or elongate) a spring is proportional to the distance the spring has been compressed (or stretched) from equilibrium; in other words, [latex]F=kx,[\/latex] where [latex]k[\/latex] is a constant<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167793609825\" class=\"definition\">\r\n \t<dt>hydrostatic pressure<\/dt>\r\n \t<dd id=\"fs-id1167793609830\">the pressure exerted by water on a submerged object<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167793609834\" class=\"definition\">\r\n \t<dt>work<\/dt>\r\n \t<dd id=\"fs-id1167793609840\">the amount of energy it takes to move an object; in physics, when a force is constant, work is expressed as the product of force and distance<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div id=\"fs-id1167791543265\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1167791543272\">\n<li>Several physical applications of the definite integral are common in engineering and physics.<\/li>\n<li>Definite integrals can be used to determine the mass of an object if its density function is known.<\/li>\n<li>Work can also be calculated from integrating a force function, or when counteracting the force of gravity, as in a pumping problem.<\/li>\n<li>Definite integrals can also be used to calculate the force exerted on an object submerged in a liquid.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1167791543292\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1167791543298\">\n<li><strong>Mass of a one-dimensional object<\/strong><br \/>\n[latex]m={\\displaystyle\\int }_{a}^{b}\\rho (x)dx[\/latex]<\/li>\n<li><strong>Mass of a circular object<\/strong><br \/>\n[latex]m={\\displaystyle\\int }_{0}^{r}2\\pi x\\rho (x)dx[\/latex]<\/li>\n<li><strong>Work done on an object<\/strong><br \/>\n[latex]W={\\displaystyle\\int }_{a}^{b}F(x)dx[\/latex]<\/li>\n<li><strong>Hydrostatic force on a plate<\/strong><br \/>\n[latex]F={\\displaystyle\\int }_{a}^{b}\\rho w(x)s(x)dx[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1167793976005\" class=\"definition\">\n<dt>density function<\/dt>\n<dd id=\"fs-id1167793609788\">a density function describes how mass is distributed throughout an object; it can be a linear density, expressed in terms of mass per unit length; an area density, expressed in terms of mass per unit area; or a volume density, expressed in terms of mass per unit volume; weight-density is also used to describe weight (rather than mass) per unit volume<\/dd>\n<\/dl>\n<dl id=\"fs-id1167793609795\" class=\"definition\">\n<dt>Hooke\u2019s law<\/dt>\n<dd id=\"fs-id1167793609800\">this law states that the force required to compress (or elongate) a spring is proportional to the distance the spring has been compressed (or stretched) from equilibrium; in other words, [latex]F=kx,[\/latex] where [latex]k[\/latex] is a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1167793609825\" class=\"definition\">\n<dt>hydrostatic pressure<\/dt>\n<dd id=\"fs-id1167793609830\">the pressure exerted by water on a submerged object<\/dd>\n<\/dl>\n<dl id=\"fs-id1167793609834\" class=\"definition\">\n<dt>work<\/dt>\n<dd id=\"fs-id1167793609840\">the amount of energy it takes to move an object; in physics, when a force is constant, work is expressed as the product of force and distance<\/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-1487\">\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 1. <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\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/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-1\/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":17533,"menu_order":21,"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\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1487","chapter","type-chapter","status-publish","hentry"],"part":65,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1487","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1487\/revisions"}],"predecessor-version":[{"id":1488,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1487\/revisions\/1488"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/65"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1487\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1487"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1487"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1487"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1487"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}