{"id":4407,"date":"2022-06-13T07:06:34","date_gmt":"2022-06-13T07:06:34","guid":{"rendered":"https:\/\/mdr.foobrdigital.com\/?p=4407"},"modified":"2022-06-13T07:06:34","modified_gmt":"2022-06-13T07:06:34","slug":"fundamental-theorems-of-integral-calculus","status":"publish","type":"post","link":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/2022\/06\/13\/fundamental-theorems-of-integral-calculus\/","title":{"rendered":"Fundamental Theorems of Integral Calculus"},"content":{"rendered":"\n<p>We define integrals as the function of the area bounded by the curve y = f(x), a \u2264 x \u2264 b, the x-axis, and the ordinates x = a and x =b, where b>a. Let x be a given point in [a,b]. Then\u00a0<\/p>\n\n\n\n<p><sup>b<\/sup>\u222b<sub>a<\/sub> f(x)dx  \u222babf(x)dx\u00a0represents the area function. This concept of area function leads to the fundamental theorems of integral calculus.<\/p>\n\n\n\n<ul><li><strong>First Fundamental Theorem of Integral Calculus<\/strong><\/li><li><strong>Second Fundamental Theorem of Integral Calculus<\/strong><\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">First Fundamental Theorem of Integrals<\/h2>\n\n\n\n<p>A(x) =\u00a0b\u222baf(x)dx\u222babf(x)dx\u00a0for all x \u2265 a,<\/p>\n\n\n\n<p> where the function is continuous on [a,b]. <\/p>\n\n\n\n<p>Then A'(x) = f(x) for all x \u03f5 [a,b]<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Second Fundamental Theorem of Integrals<\/h2>\n\n\n\n<p>If f is continuous function of x defined on the closed interval [a,b] and F be another function such that d\/dx <\/p>\n\n\n\n<p>F(x) = f(x) for all x in the domain of f, <\/p>\n\n\n\n<p>then\u00a0<sup>b<\/sup>\u222b<sub>a<\/sub>f(x)dx \u222babf(x)dx\u00a0= f(b) -f(a). <\/p>\n\n\n\n<p>This is known as the\u00a0definite integral\u00a0of f over the range [a,b], a being the lower limit and b the upper limit.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We define integrals as the function of the area bounded by the curve y = f(x), a \u2264 x \u2264 b, the x-axis, and the ordinates x = a and x =b, where b>a. Let x be a given point in [a,b]. Then\u00a0 b\u222ba f(x)dx \u222babf(x)dx\u00a0represents the area function. This concept of area function leads [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[865],"tags":[],"_links":{"self":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/4407"}],"collection":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/comments?post=4407"}],"version-history":[{"count":0,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/4407\/revisions"}],"wp:attachment":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/media?parent=4407"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/categories?post=4407"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/tags?post=4407"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}