{"id":5,"date":"2024-04-16T20:37:15","date_gmt":"2024-04-17T00:37:15","guid":{"rendered":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/?p=5"},"modified":"2024-11-03T21:23:04","modified_gmt":"2024-11-04T02:23:04","slug":"chapter-1","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/chapter\/chapter-1\/","title":{"raw":"Unit 1: Introduction to Integrals","rendered":"Unit 1: Introduction to Integrals"},"content":{"raw":"<div class=\"flex-col gap-1 md:gap-3\">\r\n<div class=\"flex flex-grow flex-col max-w-full\">\r\n<div data-message-author-role=\"assistant\" data-message-id=\"6a4a70e8-75e2-4836-9b19-6b3f952ca3e3\" dir=\"auto\" class=\"min-h-[20px] text-message flex flex-col items-start gap-3 whitespace-pre-wrap break-words [.text-message+&amp;]:mt-5 overflow-x-auto\">\r\n<div class=\"markdown prose w-full break-words dark:prose-invert light\">\r\n\r\nIn integral calculus, you'll encounter two primary types of integrals:\r\n\r\n<strong>indefinite integrals<\/strong>, [latex]\\int f(x) \\, dx[\/latex],\r\n\r\nand <strong>definite integrals<\/strong>, [latex]\\int_{b}^{a} f(x) \\, dx[\/latex].\r\n\r\nIndefinite integrals represent a family of functions rather than a specific value. They are expressed using the integral symbol without upper and lower limits. Indefinite integrals essentially answer the question \"What function has a derivative equal to the integrand?\"\r\n\r\nOn the other hand, definite integrals have specific bounds, indicating the range over which the integration is performed. They yield a single numerical value, representing the accumulated area under the curve between the specified limits.\r\n\r\nThe main difference lies in their outcomes: indefinite integrals produce functions, while definite integrals yield numerical values. For example\r\n<p style=\"text-align: center\">[latex]\\int 3x^2 \\, dx = x^3+C, \\,\\,\\,\\,\\,\\, \\int_{1}^2 3x^2 \\, dx = 7.\u00a0 [\/latex]<\/p>\r\nWe will learn the techniques how to solve indefinite and definite integrals.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"flex-col gap-1 md:gap-3\">\n<div class=\"flex flex-grow flex-col max-w-full\">\n<div data-message-author-role=\"assistant\" data-message-id=\"6a4a70e8-75e2-4836-9b19-6b3f952ca3e3\" dir=\"auto\" class=\"min-h-[20px] text-message flex flex-col items-start gap-3 whitespace-pre-wrap break-words [.text-message+&amp;]:mt-5 overflow-x-auto\">\n<div class=\"markdown prose w-full break-words dark:prose-invert light\">\n<p>In integral calculus, you&#8217;ll encounter two primary types of integrals:<\/p>\n<p><strong>indefinite integrals<\/strong>, [latex]\\int f(x) \\, dx[\/latex],<\/p>\n<p>and <strong>definite integrals<\/strong>, [latex]\\int_{b}^{a} f(x) \\, dx[\/latex].<\/p>\n<p>Indefinite integrals represent a family of functions rather than a specific value. They are expressed using the integral symbol without upper and lower limits. Indefinite integrals essentially answer the question &#8220;What function has a derivative equal to the integrand?&#8221;<\/p>\n<p>On the other hand, definite integrals have specific bounds, indicating the range over which the integration is performed. They yield a single numerical value, representing the accumulated area under the curve between the specified limits.<\/p>\n<p>The main difference lies in their outcomes: indefinite integrals produce functions, while definite integrals yield numerical values. For example<\/p>\n<p style=\"text-align: center\">[latex]\\int 3x^2 \\, dx = x^3+C, \\,\\,\\,\\,\\,\\, \\int_{1}^2 3x^2 \\, dx = 7.\u00a0[\/latex]<\/p>\n<p>We will learn the techniques how to solve indefinite and definite integrals.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":391,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[48],"contributor":[],"license":[],"class_list":["post-5","chapter","type-chapter","status-publish","hentry","chapter-type-standard"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/5","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/users\/391"}],"version-history":[{"count":6,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/5\/revisions"}],"predecessor-version":[{"id":256,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/5\/revisions\/256"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapters\/5\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/media?parent=5"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/pressbooks\/v2\/chapter-type?post=5"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/contributor?post=5"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/videointegrals\/wp-json\/wp\/v2\/license?post=5"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}