{"id":133896,"date":"2023-03-11T17:22:21","date_gmt":"2023-03-11T11:52:21","guid":{"rendered":"https:\/\/www.mapsofindia.com\/my-india\/?p=133896"},"modified":"2023-03-11T17:22:21","modified_gmt":"2023-03-11T11:52:21","slug":"sketch-the-region-bounded-by-the-lines-2x-y-8-y-2-y-4-and-the-y-axis-hence-obtain-its-area-using-integration","status":"publish","type":"post","link":"https:\/\/www.mapsofindia.com\/my-india\/quiz\/sketch-the-region-bounded-by-the-lines-2x-y-8-y-2-y-4-and-the-y-axis-hence-obtain-its-area-using-integration","title":{"rendered":"Sketch the region bounded by the lines 2x + y = 8, y = 2, y = 4 and the y-axis. Hence, obtain its area using integration."},"content":{"rendered":"<h2>Question: Sketch the region bounded by the lines 2x + y = 8, y = 2, y = 4 and the y-axis. Hence, obtain its area using integration.<\/h2>\n<h3>The correct answer is &#8211;<\/h3>\n<p>To sketch the region, we start by plotting the lines on a coordinate axis.<\/p>\n<p>First, we can find the x-intercept of the line 2x + y = 8 by setting y = 0:<\/p>\n<p>2x + 0 = 8<\/p>\n<p>x = 4<\/p>\n<p>So, the line passes through the point (4, 0).<\/p>\n<p>Next, we can find the points where the line intersects the other two given lines:<\/p>\n<ul>\n<li>When y = 2:<\/li>\n<\/ul>\n<p>2x + y = 8<\/p>\n<p>2x + 2 = 8<\/p>\n<p>x = 3<\/p>\n<p>So, the line passes through the point (3, 2).<\/p>\n<ul>\n<li>When y = 4:<\/li>\n<\/ul>\n<p>2x + y = 8<\/p>\n<p>2x + 4 = 8<\/p>\n<p>x = 2<\/p>\n<p>So, the line passes through the point (2, 4).<\/p>\n<p>We can now plot the lines and shade the region bounded by them and the y-axis:<\/p>\n<p>&nbsp;<\/p>\n<p>|<br \/>\n4 +&#8212;&#8212;&#8212;&#8212;&#8211;+<br \/>\n| |<br \/>\n3 +&#8212;&#8212;&#8212;&#8212;&#8211;+<br \/>\n| Region |<br \/>\n2 +&#8212;&#8212;+ |<br \/>\n| | |<br \/>\n1 +&#8212;&#8212;+&#8212;&#8212;-+<br \/>\n| | | | |<br \/>\n| | | | |<br \/>\n| | | | |<br \/>\n| | | | |<br \/>\n| | | | |<br \/>\n| | | | |<br \/>\n+&#8211;+&#8212;+&#8212;+&#8212;+<br \/>\n0 2 3 4<\/p>\n<div class=\"group w-full text-gray-800 dark:text-gray-100 border-b border-black\/10 dark:border-gray-900\/50 bg-gray-50 dark:bg-[#444654]\">\n<div class=\"text-base gap-4 md:gap-6 md:max-w-2xl lg:max-w-2xl xl:max-w-3xl p-4 md:py-6 flex lg:px-0 m-auto\">\n<div class=\"relative flex w-[calc(100%-50px)] flex-col gap-1 md:gap-3 lg:w-[calc(100%-115px)]\">\n<div class=\"flex flex-grow flex-col gap-3\">\n<div class=\"min-h-[20px] flex flex-col items-start gap-4 whitespace-pre-wrap\">\n<div class=\"markdown prose w-full break-words dark:prose-invert light\">\n<p>To find the area of this region, we can integrate the area of the vertical strips that make up the region.<\/p>\n<p>The strips are bounded by the y-axis on one side and the line 2x + y = 8 on the other side. We can express this line in terms of x as:<\/p>\n<p>y = 8 &#8211; 2x<\/p>\n<p>So the height of each strip is given by the difference between the y-coordinate of the line and the y-coordinate of the y-axis (which is 0).<\/p>\n<p>The width of each strip is dx.<\/p>\n<p>Therefore, the area of each strip is:<\/p>\n<p>dA = (8 &#8211; 2x) dx<\/p>\n<p>To find the total area, we integrate this expression over the range of x values that define the region:<\/p>\n<p>A = \u222b(from x=0 to x=2) (8 &#8211; 2x) dx + \u222b(from x=2 to x=3) (4 &#8211; 2x) dx + \u222b(from x=3 to x=4) (2) dx<\/p>\n<p>Simplifying this expression, we get:<\/p>\n<p>A = [8x &#8211; x^2] from x=0 to x=2 + [4x &#8211; x^2] from x=2 to x=3 + [2x] from x=3 to x=4<\/p>\n<p>A = 12<\/p>\n<p>Therefore, the area of the region bounded by the lines 2x + y = 8, y = 2, y = 4 and the y-axis is 12 square units.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"flex justify-between\">\n<div class=\"text-gray-400 flex self-end lg:self-center justify-center mt-2 gap-3 md:gap-4 lg:gap-1 lg:absolute lg:top-0 lg:translate-x-full lg:right-0 lg:mt-0 lg:pl-2 visible\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Question: Sketch the region bounded by the lines 2x + y = 8, y = 2, y = 4 and the y-axis. Hence, obtain its area using integration. The correct answer is &#8211; To sketch the region, we start by plotting the lines on a coordinate axis. First, we can find the x-intercept of the [&hellip;]<\/p>\n","protected":false},"author":21842,"featured_media":132233,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12119],"tags":[],"class_list":{"0":"post-133896","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-quiz"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/133896","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/users\/21842"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/comments?post=133896"}],"version-history":[{"count":1,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/133896\/revisions"}],"predecessor-version":[{"id":133897,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/133896\/revisions\/133897"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media\/132233"}],"wp:attachment":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media?parent=133896"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/categories?post=133896"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/tags?post=133896"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}