{"id":250816,"date":"2025-07-10T12:30:01","date_gmt":"2025-07-10T12:30:01","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=250816"},"modified":"2025-07-10T12:30:05","modified_gmt":"2025-07-10T12:30:05","slug":"what-is-the-difference-between-net-area-and-total-area-from-the-curve-to-the-x-axis","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/10\/what-is-the-difference-between-net-area-and-total-area-from-the-curve-to-the-x-axis\/","title":{"rendered":"What is the difference between net area and total area from the curve to the x-axis"},"content":{"rendered":"\n<p>What is the difference between net area and total area from the curve to the x-axis? What is the difference in using definite integrals to calculate the total area from the curve to the x-axis from x = a to x = b rather than the net area? What would the solutions to integral(s) calculating the net area as well as total area of a velocity function tell you about the situation? Include examples in your detailed explanations.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The key difference between <strong>net area<\/strong> and <strong>total area<\/strong> from the curve to the x-axis lies in how areas above and below the x-axis are treated.<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Net Area<\/strong>:<br>The <strong>net area<\/strong> refers to the algebraic sum of the areas between the curve and the x-axis over a given interval [a, b]. It takes into account both positive and negative areas, depending on whether the curve is above or below the x-axis. Areas above the x-axis are positive, while areas below the x-axis are negative. Mathematically, this is calculated using a <strong>definite integral<\/strong>: Net\u00a0Area=\u222babf(x)\u2009dx\\text{Net Area} = \\int_a^b f(x) \\, dxNet\u00a0Area=\u222bab\u200bf(x)dx Example:<br>For f(x)=xf(x) = xf(x)=x from a=\u22121a = -1a=\u22121 to b=1b = 1b=1: \u222b\u221211x\u2009dx=0\\int_{-1}^1 x \\, dx = 0\u222b\u221211\u200bxdx=0 This is because the areas above and below the x-axis cancel out.<\/li>\n\n\n\n<li><strong>Total Area<\/strong>:<br>The <strong>total area<\/strong> refers to the sum of the absolute areas between the curve and the x-axis, regardless of whether the curve is above or below the x-axis. In this case, negative areas are treated as positive. To find the total area, you would compute the integral of the <strong>absolute value<\/strong> of the function: Total\u00a0Area=\u222bab\u2223f(x)\u2223\u2009dx\\text{Total Area} = \\int_a^b |f(x)| \\, dxTotal\u00a0Area=\u222bab\u200b\u2223f(x)\u2223dx Example:<br>For f(x)=xf(x) = xf(x)=x from a=\u22121a = -1a=\u22121 to b=1b = 1b=1, the total area is: \u222b\u221211\u2223x\u2223\u2009dx=2(since\u00a0the\u00a0area\u00a0above\u00a0and\u00a0below\u00a0the\u00a0x-axis\u00a0is\u00a0treated\u00a0as\u00a0positive)\\int_{-1}^1 |x| \\, dx = 2 \\quad (\\text{since the area above and below the x-axis is treated as positive})\u222b\u221211\u200b\u2223x\u2223dx=2(since\u00a0the\u00a0area\u00a0above\u00a0and\u00a0below\u00a0the\u00a0x-axis\u00a0is\u00a0treated\u00a0as\u00a0positive)<\/li>\n\n\n\n<li><strong>Definite Integrals for Net and Total Areas of a Velocity Function<\/strong>:<br>When calculating the net area under a velocity function (say, v(t)v(t)v(t) over a time interval), the <strong>net area<\/strong> would give the <strong>displacement<\/strong>, which accounts for both the motion in the positive and negative directions. If v(t)v(t)v(t) is positive, the object moves in one direction, and if v(t)v(t)v(t) is negative, the object moves in the opposite direction. The <strong>total area<\/strong> would give the <strong>total distance traveled<\/strong>, irrespective of direction, because all negative velocities are considered as positive. Example:<br>For a velocity function v(t)=5\u2212t2v(t) = 5 &#8211; t^2v(t)=5\u2212t2 over the interval [0,2][0, 2][0,2]:<ul><li>Net area (displacement):<\/li><\/ul>\u222b02(5\u2212t2)\u2009dt=[5t\u2212t33]02=(10\u221283)\u22120=303\u221283=223\u22487.33\\int_0^2 (5 &#8211; t^2) \\, dt = \\left[ 5t &#8211; \\frac{t^3}{3} \\right]_0^2 = (10 &#8211; \\frac{8}{3}) &#8211; 0 = \\frac{30}{3} &#8211; \\frac{8}{3} = \\frac{22}{3} \\approx 7.33\u222b02\u200b(5\u2212t2)dt=[5t\u22123t3\u200b]02\u200b=(10\u221238\u200b)\u22120=330\u200b\u221238\u200b=322\u200b\u22487.33<ul><li>Total area (total distance traveled):<\/li><\/ul>\u222b02\u22235\u2212t2\u2223\u2009dt\\int_0^2 |5 &#8211; t^2| \\, dt\u222b02\u200b\u22235\u2212t2\u2223dt (This would require splitting the integral at points where 5\u2212t25 &#8211; t^25\u2212t2 changes sign, but essentially the result gives the total distance.)<\/li>\n<\/ol>\n\n\n\n<p>In summary:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Net area<\/strong> accounts for direction, leading to displacement.<\/li>\n\n\n\n<li><strong>Total area<\/strong> treats all movement as positive, giving total distance traveled.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-154.jpeg\" alt=\"\" class=\"wp-image-250817\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>What is the difference between net area and total area from the curve to the x-axis? What is the difference in using definite integrals to calculate the total area from the curve to the x-axis from x = a to x = b rather than the net area? What would the solutions to integral(s) calculating [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","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":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-250816","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/250816","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=250816"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/250816\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=250816"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=250816"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=250816"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}