{"id":29768,"date":"2022-05-15T13:12:29","date_gmt":"2022-05-15T07:42:29","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=29768"},"modified":"2024-05-15T20:20:03","modified_gmt":"2024-05-15T14:50:03","slug":"area-and-volume","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=29768","title":{"rendered":"Area and Volume"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[We\\ apply\\ the\\ concept\\ of\\ definite\\ integral\\ to\\ find\\ the\\ area\\ and\\ volume\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading has-vivid-cyan-blue-color has-text-color\" style=\"font-size:18px\">Area<\/h2>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ area\\ under\\ the\\ curve\\ y\\ =\\ f(x)\\ between\\ the\\ x-axis\\ and\\]\\[the\\ ordinates\\ x\\ =\\ a\\ and\\ x\\ =\\ b\\ is\\ given\\ by\\]\\[\\boxed{Area =\\    \\int_{a}^{b} y\\ dx}\\]<\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img data-recalc-dims=\"1\" fetchpriority=\"high\" decoding=\"async\" width=\"493\" height=\"204\" data-attachment-id=\"29772\" data-permalink=\"https:\/\/yanamtakshashila.com\/?attachment_id=29772\" data-orig-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/image.png?fit=493%2C204&amp;ssl=1\" data-orig-size=\"493,204\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"image\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/image.png?fit=493%2C204&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/image.png?resize=493%2C204&#038;ssl=1\" alt=\"\" class=\"wp-image-29772\"\/><\/figure>\n\n\n\n<p class=\"has-vivid-cyan-blue-color has-text-color\" style=\"font-size:18px\">Volume<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ volume\\ of\\ the\\ solid\\ obtained\\ by\\ rotating\\ the\\ area\\  bounded\\ by\\ the\\ curve\\ y\\ =\\ f(x)\\, the\\ x \u2013 axis\\]\\[and\\ the\\  ordinates\\ is\\ given\\ by\\]\\[\\boxed{Volume\\  =\\ \u03c0  \\int_{a}^{b} y^2\\ dx}\\]<\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img data-recalc-dims=\"1\" decoding=\"async\" width=\"563\" height=\"306\" data-attachment-id=\"29778\" data-permalink=\"https:\/\/yanamtakshashila.com\/?attachment_id=29778\" data-orig-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/image-1.png?fit=563%2C306&amp;ssl=1\" data-orig-size=\"563,306\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"image-1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/image-1.png?fit=563%2C306&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/image-1.png?resize=563%2C306&#038;ssl=1\" alt=\"\" class=\"wp-image-29778\"\/><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 1\\ .}\\ \\color {red}{Find\\ the\\ Area}\\ bounded\\ by\\ the\\ curve\\ y\\ =\\ f(x)\\ \\hspace{8cm}\\]\\[the\\ X-axis\\ and\\ ordinates\\ x\\ =\\ a\\ and\\ x\\ =\\ b\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ y\\ =\\ f(x)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Area =\\    \\int_{a}^{b} y\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Area =\\    \\int_{a}^{b} f(x)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 2\\ .}\\ \\color {red}{Find\\ the\\ Area}\\ bounded\\ by\\ the\\ curve\\ y\\ =\\ x^2\\ \\hspace{8cm}\\]\\[the\\ X-axis\\ and\\ ordinates\\ x\\ =\\ 0\\ and\\ x\\ =\\ 2\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ y\\ =\\ x^2,\\ a\\ =\\ 0\\ and\\ b\\ =\\ 2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Area =\\    \\int_{a}^{b} y\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Area =\\    \\int_{0}^{2} x^2\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{x^3}{3} \\Biggr]_{0}^{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= [\\frac{2^3}{3} &#8211; \\frac{0^3}{3}]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{8}{3} &#8211; \\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{8}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Area\\ = \\frac{8}{3}}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/RvZ_dWTmCZM\" title=\"Area and Volume (Application of Integration) - Part - 2\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3\\ .}\\ \\color {red}{Find\\ the\\ Area}\\ bounded\\ by\\ the\\ curve\\ y\\ =\\ x^2\\ +\\ x\\ \\hspace{8cm}\\]\\[the\\ X-axis\\ between\\ x\\ =\\ 0\\ and\\ x\\ =\\ 4\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ y\\ =\\ x^2\\ +\\ x,\\ a\\ =\\ 0\\ and\\ b\\ =\\ 4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Area =\\    \\int_{a}^{b} y\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Area =\\    \\int_{0}^{4} (x^2\\ +\\ x)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{x^3}{3}\\ +\\ \\frac{x^2}{2} \\Biggr]_{0}^{4}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= [(\\frac{4^3}{3}\\ +\\ \\frac{4^2}{2})\\ -\\ (\\frac{0^3}{3}\\ +\\ \\frac{0^2}{2}) ]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{64}{3}\\ + \\ 8\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{64\\ +\\ 24}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{88}{3}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/laSgzXltdn4\" title=\"Area and Volume - Part - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 4\\ .}\\ \\color {red}{Find\\ the\\ Volume}\\ of\\ the\\ solid\\ formed\\ when\\ the\\ area\\ bounded\\ by\\ the\\ curve\\ y^2\\ =\\ 4\\  x\\ \\hspace{8cm}\\]\\[between\\ x\\ =\\ 0\\ and\\ x\\ =\\ 1\\ is\\ rotated\\ about\\ the\\ X\\ -\\ axis.\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ y^2\\ =\\ 4\\ x,\\ a\\ =\\ 0\\ and\\ b\\ =\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Volume\\  =\\ \u03c0  \\int_{a}^{b} y^2\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \u03c0  \\int_{0}^{1} 4\\ x\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= 4\\  \\pi\\ \\Biggr[\\frac{x^2}{2} \\Biggr]_{0}^{1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ 4\\  \\pi  \\Biggr[ \\frac{1^2}{2}\\ -\\ \\frac{0^2}{2} \\Biggr]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ 4\\ \\pi (\\frac{1} {2})\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ 2\\  \\pi\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Volume\\ =\\ 2\\ \\pi\\ cubic\\ units}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/oEJrdP5-WU4\" title=\"Area and Volume (Application of Integration) - Part - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 5\\ .}\\ \\color {red}{Find\\ the\\ Volume}\\  bounded\\ by\\ the\\ curve\\ y^2\\ =\\ x^6,\\ \\hspace{8cm}\\]\\[the\\ X\\ -\\ axis\\ and\\ ordinates\\ x\\ =\\ 0\\ and\\ x\\ =\\ 1\\ \\hspace{3cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Given\\ y^2\\ =\\ x^6,\\ a\\ =\\ 0\\ and\\ b\\ =\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Volume\\  =\\ \u03c0  \\int_{a}^{b} y^2\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \u03c0  \\int_{0}^{1} x^6\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \u03c0  \\frac{x^7}{7} \\Biggr]_{0}^{1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \u03c0 [\\frac{1^7}{7} &#8211; \\frac{0^7}{7}]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{\u03c0}{7}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Volume\\ = \\frac{\u03c0}{7}}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/xWXQ9PfDNVA\" title=\"Area and Volume (Application of Integration) - Part - 5\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"6771622407\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<p class=\"has-vivid-purple-color has-text-color\" style=\"font-size:15px\"><strong>Example &nbsp;4:<\/strong><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red}{Find\\ the\\ Area\\ bounded\\ by\\ the\\ curve\\ y\\ =\\ x^2\\ -\\ 6x\\ +\\ 8}\\ \\hspace{10cm}\\]\\[\\color{red}{and\\ the\\ X-axis}\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[To\\ find\\ limits\\ put\\ y\\ =\\ 0\\ as\\ the\\ curve\\ meets\\ X-\\ axis\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2\\ -\\ 6x\\ +\\ 8\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2\\ -\\ 2x\\ -\\ 4x +\\ 8\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x(x\\ -\\ 2)\\ -\\ 4(x\\ -\\ 2)\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(x\\ -\\ 2)(x\\ -\\ 4)\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ limts\\ are\\ x\\ =\\ \\ 2\\ and\\ x\\ =\\ 4\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Area =\\    \\int_{a}^{b} y\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Area\\  =\\   \\int_{2}^{4} (x^2\\ -\\ 6x\\ +\\ 8)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\  \\Biggr[\\frac{x^3}{3}\\ -\\ 6\\ \\frac{x^2}{2}\\ +\\ 8\\ x \\Biggr]_{2}^{4}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ \\Biggr[(\\frac{4^3}{3}\\ -\\ 6\\frac{4^2}{2}\\ +\\  8(4))\\ -\\ (\\frac{(2)^3}{3}\\ -\\ 6\\frac{2^2}{2}\\ +\\  8(2))\\Biggr]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ \\Biggr[(\\frac{64}{3}\\ -\\ 48\\ +\\ 32)\\ -\\ (\\frac{8}{3}\\ -\\ 12\\ +\\  16)\\Biggr]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ \\Biggr[(\\frac{64}{3}\\ -\\ 16)\\ -\\ (\\frac{8}{3}\\ +\\ 4)\\Biggr]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ \\Biggr[\\frac{64}{3}\\ -\\ \\frac{8}{3}\\ -\\ 20\\Biggr]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ \\Biggr[\\frac{64\\ -\\ 8\\ -\\ 60}{3}\\Biggr]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\frac{-\\ 4}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Area\\ =\\ \\frac{4}{3}\\ sq\\ units (as\\ Area\\ is\\ positive)}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/suB7aGHewVY\" title=\"Area and Volume (Application of Integration) - Part - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<p class=\"has-vivid-purple-color has-text-color\" style=\"font-size:15px\"><strong>Example &nbsp;5:<\/strong><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{red}{Find\\ the\\ Area\\ of\\ the\\ circle\\ whose\\ radius\\ is\\ &#8216;a&#8217;\\ units}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img data-recalc-dims=\"1\" decoding=\"async\" width=\"840\" height=\"603\" data-attachment-id=\"29822\" data-permalink=\"https:\/\/yanamtakshashila.com\/?attachment_id=29822\" data-orig-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-166-1.jpg?fit=1034%2C742&amp;ssl=1\" data-orig-size=\"1034,742\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Engineering-Mathematics-II-166-1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-166-1.jpg?fit=840%2C603&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-166-1.jpg?resize=840%2C603&#038;ssl=1\" alt=\"\" class=\"wp-image-29822\" style=\"width:397px;height:285px\" srcset=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-166-1.jpg?resize=840%2C603&amp;ssl=1 840w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-166-1.jpg?resize=509%2C365&amp;ssl=1 509w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-166-1.jpg?resize=768%2C551&amp;ssl=1 768w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-166-1.jpg?resize=240%2C172&amp;ssl=1 240w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-166-1.jpg?w=1034&amp;ssl=1 1034w\" sizes=\"(max-width: 840px) 100vw, 840px\" \/><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Equation\\ of\\ circle\\ is\\ x^2\\ +\\ y^2\\ =\\ a^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y^2\\ =\\ a^2\\ -\\ x^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y\\ =\\ \\sqrt{a^2\\ -\\ x^2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Area\\ of\\ OAB\\ =\\ \\int_{0}^{a} \\sqrt{a^2\\ -\\ x^2}\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[put\\ x\\ =\\ a\\ sin\\ \\theta\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dx}{d\\theta}\\ =\\ a\\ cos\\ \\theta\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[dx\\ =\\ a\\ cos\\ \\theta\\ d\\theta\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[when\\ x\\ =\\ 0\\ \\hspace{2cm}\\ \\theta\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[when\\ x\\ =\\ a\\ \\hspace{2cm}\\ a\\ =\\ a\\ sin\\ \\theta\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{4cm}\\ sin\\ \\theta\\ =\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{4cm}\\ \\theta\\ =\\ \\frac{\\pi}{2}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Area\\ of\\ OAB\\ =\\ \\int_{0}^{\\frac{\\pi}{2}} \\sqrt{a^2\\ -\\ a^2\\ sin^2\\ \\theta}\\ a\\ cos\\ \\theta\\ d \\theta\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\int_{0}^{\\frac{\\pi}{2}}\\ a \\sqrt{1\\ -\\  sin^2\\ \\theta}\\ a\\ cos\\ \\theta\\ d \\theta\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\int_{0}^{\\frac{\\pi}{2}}\\ a \\ cos\\ \\theta\\ a\\ cos\\ \\theta\\ d \\theta\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\int_{0}^{\\frac{\\pi}{2}}\\ a^2 \\ cos^2\\ \\theta\\  d \\theta\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\int_{0}^{\\frac{\\pi}{2}}\\ a^2 \\ (\\frac{1\\ +\\ cos\\ 2\\ \\theta}{2})\\  d \\theta\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{a^2}{2}[\\int_{0}^{\\frac{\\pi}{2}} 1\\ d \\theta\\ +\\ \\int_{0}^{\\frac{\\pi}{2}} cos\\ 2\\ \\theta\\  d \\theta]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{a^2}{2}\\ [ \\Biggr[ \\theta \\Biggr]_{0}^{\\frac{\\pi}{2}}\\ +\\ \\Biggr[\\frac{sin\\ 2\\theta}{2} \\Biggr]_{0}^{\\frac{\\pi}{2}}]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{a^2}{2}\\ [(\\frac{\\pi}{2}\\ -\\ 0)\\ +\\ \\frac{1}{2}(sin\\ \\pi\\ -\\ sin\\ 0)]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\frac{a^2}{2}\\ [(\\frac{\\pi}{2})\\ +\\ \\frac{1}{2}(0)]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Area\\ of\\ OAB\\ =\\ \\frac{\\pi\\ a^2}{4}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Area\\ of\\ circle\\ =\\ 4\\ \\times\\ Area\\ of\\ OAB\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 4\\ \\times\\ \\frac{\\pi\\ a^2}{4}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\pi\\ a^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Area\\ of\\ circle\\ =\\ \\pi\\ a^2}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/U-2bZFx8hQo\" title=\"Area and Volume (Application of Integration) Part - 5\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<p class=\"has-vivid-purple-color has-text-color\" style=\"font-size:15px\"><strong>Example &nbsp;6:<\/strong><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{red}{Find\\ the\\ volume\\ of\\  sphere\\ having\\ radius\\ &#8216;a&#8217;\\ by\\ using\\ integration}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"840\" height=\"479\" data-attachment-id=\"29825\" data-permalink=\"https:\/\/yanamtakshashila.com\/?attachment_id=29825\" data-orig-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169.jpg?fit=1168%2C666&amp;ssl=1\" data-orig-size=\"1168,666\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Engineering-Mathematics-II-169\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169.jpg?fit=840%2C479&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169.jpg?resize=840%2C479&#038;ssl=1\" alt=\"\" class=\"wp-image-29825\" style=\"width:463px;height:264px\" srcset=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169.jpg?resize=840%2C479&amp;ssl=1 840w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169.jpg?resize=640%2C365&amp;ssl=1 640w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169.jpg?resize=768%2C438&amp;ssl=1 768w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169.jpg?w=1168&amp;ssl=1 1168w\" sizes=\"(max-width: 840px) 100vw, 840px\" \/><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Rotate\\ the\\ area\\ OAB\\ (Quadrant\\ of\\ circle)\\ about\\ OA,\\  the\\ x- axis.\\]\\[Then\\ we\\ get\\ a\\ hemisphere.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Volume\\ of\\ hemisphere\\ is\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[V\\  =\\ \u03c0  \\int_{0}^{a} y^2\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\ =\\ \u03c0  \\int_{0}^{a} (a^2\\ -\\ x^2)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\pi \\Biggr[a^2\\ x\\ -\\ \\frac{x^3}{3}\\Biggr]_ {0}^{a}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\pi \\Biggr[a^2\\ a\\ -\\ \\frac{a^3}{3}\\Biggr]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\pi \\Biggr[a^3\\  -\\ \\frac{a^3}{3}\\Biggr]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\pi \\Biggr[\\frac{3a^3\\ -\\ a^3}{3}\\Biggr]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \\pi \\Biggr[\\frac{2a^3}{3}\\Biggr]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[V\\  =\\ \\frac{2\\ \\pi\\ a^3}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Volume\\ of\\ sphere\\ =\\ 2\\ \\times\\ \\frac{2\\ \\pi\\ a^3}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Volume\\ of\\ sphere\\ =\\ \\frac{4\\ \\pi\\ a^3}{3}}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/rrf6BsCQF_Y\" title=\"Area and Volume (Application of Integration) Part - 6\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Leader board 1 -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:728px;height:90px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8769628924\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<p class=\"has-vivid-purple-color has-text-color\" style=\"font-size:15px\"><strong>Example &nbsp;7:<\/strong><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{red}{Find\\ the\\ volume\\ of\\ right\\ circular\\ cone\\ \\ of\\ height\\ &#8216;h&#8217;\\ and\\ base\\ radius\\ &#8216;r&#8217;\\ by\\ integration}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"840\" height=\"630\" data-attachment-id=\"29833\" data-permalink=\"https:\/\/yanamtakshashila.com\/?attachment_id=29833\" data-orig-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169-1.jpg?fit=909%2C682&amp;ssl=1\" data-orig-size=\"909,682\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Engineering-Mathematics-II-169-1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169-1.jpg?fit=840%2C630&amp;ssl=1\" src=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169-1.jpg?resize=840%2C630&#038;ssl=1\" alt=\"\" class=\"wp-image-29833\" style=\"width:311px;height:233px\" srcset=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169-1.jpg?resize=840%2C630&amp;ssl=1 840w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169-1.jpg?resize=486%2C365&amp;ssl=1 486w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169-1.jpg?resize=768%2C576&amp;ssl=1 768w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169-1.jpg?resize=800%2C600&amp;ssl=1 800w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169-1.jpg?resize=400%2C300&amp;ssl=1 400w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169-1.jpg?resize=200%2C150&amp;ssl=1 200w, https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2022\/05\/Engineering-Mathematics-II-169-1.jpg?w=909&amp;ssl=1 909w\" sizes=\"(max-width: 840px) 100vw, 840px\" \/><\/figure>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Equation\\ of\\ OB\\ is\\ y\\ =\\ mx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[m\\ =\\ tan\\ \\theta\\ =\\ \\frac{r}{h}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Volume\\ of\\ the\\ cone\\ is\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[V\\  =\\ \u03c0  \\int_{0}^{h} y^2\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[V\\  =\\ \u03c0  \\int_{0}^{h} m^2\\ x^2\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ \u03c0  \\int_{0}^{h} \\frac{r^2}{h^2}\\ x^2\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ \u03c0  \\frac{r^2}{h^2} \\Biggr[ \\frac{x^3}{3}\\ \\Biggr]_{0}^{h}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ \u03c0  \\frac{r^2}{h^2} \\Biggr[ \\frac{h^3}{3}\\ -\\ \\frac{0^3}{3} \\Biggr]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ \u03c0  \\frac{r^2}{h^2} \\frac{h^3}{3}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Volume\\ of\\ cone\\ =\\ \\frac{1}{3}\\ \\pi\\ r^2\\ h}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/Vlv5AkEbkDU\" title=\"Area and Volume ( Application of Integration)  Part - 7\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<h4 class=\"wp-block-heading has-vivid-purple-color has-text-color\"><strong>Example &nbsp;8:<\/strong><\/h4>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red}{Find\\ the\\ Volume\\ of\\ the\\ solid\\ generated\\ by\\ the\\ area\\ enclosed\\ by\\ the\\ curve\\ y^2\\ =\\ x(x\\ -\\ 1)^2}\\ \\hspace{10cm}\\]\\[\\color{red}{and\\ the\\ X-axis\\ when\\ rotated\\ about\\ X- axis}\\  \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace {18cm}\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Soln:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y^2\\ =\\ x(x\\ -\\ 1)^2\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[To\\ find\\ limits\\ put\\ y\\ =\\ 0\\ as\\ the\\ curve\\ meets\\ X-\\ axis\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x(x\\ -\\ 1)^2\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x\\ =\\ 0\\ and\\ (x\\ -\\ 1)^2\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ limts\\ are\\ x\\ =\\  0\\ and\\ x\\ =\\ 1\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Volume\\  =\\ \u03c0  \\int_{a}^{b} y^2\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[V\\  =\\ \u03c0  \\int_{0}^{1} x(x\\ -\\ 1)^2\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ \u03c0  \\int_{0}^{1} x (x^2\\ -\\ 2x\\ +\\ 1)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \u03c0  \\int_{0}^{1}  (x^3\\ -\\ 2x^2\\ +\\ x)\\ dx\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \u03c0 \\Biggr[\\frac{x^4}{4}\\ -\\ 2\\ \\frac{x^3}{3}\\ +\\ \\frac{x^2}{2} \\Biggr]_{0}^{1}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"> \\[=\\ \u03c0 \\Biggr[(\\frac{1^4}{4}\\ -\\ 2\\ \\frac{1^3}{3}\\ +\\ \\frac{1^2}{2})\\ -\\ ((\\frac{0^4}{4}\\ -\\ 2\\ \\frac{0^3}{3}\\ +\\ \\frac{0^2}{2})\\Biggr]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \u03c0 \\Biggr[(\\frac{1}{4}\\ -\\ \\frac{2}{3}\\ +\\ \\frac{1}{2})\\ -\\ (0\\ -\\ 2(0)\\ +\\ 0)\\Biggr]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \u03c0 \\Biggr[(\\frac{3\\ -\\ 8\\ +\\ 6}{12})\\ -\\ (0) \\Biggr]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ \u03c0 \\Biggr[\\frac{1}{12}\\ \\Biggr]\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Volume\\ =\\  \\frac{\u03c0}{12}\\ cubic\\ units}\\]<\/div>\n\n\n<p><iframe width=\"790\" height=\"444\" src=\"https:\/\/www.youtube.com\/embed\/45tpjIaLJno\" title=\"Area and Volume (Application of Integration) - Part - 8\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Area Volume Example &nbsp;4: Example &nbsp;5: Example &nbsp;6: Example &nbsp;7: Example &nbsp;8:<\/p>\n","protected":false},"author":187055548,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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