{"id":230328,"date":"2025-06-09T06:58:47","date_gmt":"2025-06-09T06:58:47","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=230328"},"modified":"2025-06-09T06:58:49","modified_gmt":"2025-06-09T06:58:49","slug":"the-value-of-a-definite-integral-will-always-be-positive","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/09\/the-value-of-a-definite-integral-will-always-be-positive\/","title":{"rendered":"The value of a definite integral will always be positive"},"content":{"rendered":"\n

The value of a definite integral will always be positive. ? True ? False<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Correct Answer: \u274c False<\/strong><\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

The value of a definite integral is not always positive<\/strong>. A definite integral represents the net area<\/strong> between the graph of a function and the x-axis over a given interval [a,b][a, b][a,b]. This net area can be positive, negative, or zero<\/strong>, depending on the behavior of the function within that interval.<\/p>\n\n\n\n

Positive vs. Negative Area:<\/h4>\n\n\n\n
    \n
  • If the function f(x)f(x)f(x) is entirely above the x-axis<\/strong> on the interval [a,b][a, b][a,b] (i.e., f(x)\u22650f(x) \\geq 0f(x)\u22650 for all xxx in that interval), then the definite integral \u222babf(x)\u2009dx\\int_a^b f(x) \\, dx\u222bab\u200bf(x)dx will be positive or zero<\/strong>.<\/li>\n\n\n\n
  • If the function f(x)f(x)f(x) is entirely below the x-axis<\/strong> on the interval (i.e., f(x)\u22640f(x) \\leq 0f(x)\u22640), then the integral will be negative or zero<\/strong>.<\/li>\n\n\n\n
  • If the function crosses the x-axis (i.e., takes on both positive and negative values), the integral adds up positive and negative contributions, resulting in the net area<\/strong>, which could be positive, negative, or zero<\/strong>.<\/li>\n<\/ul>\n\n\n\n

    Example:<\/h4>\n\n\n\n

    Consider the function f(x)=\u22122f(x) = -2f(x)=\u22122 over the interval [0,3][0, 3][0,3]:\u222b03\u22122\u2009dx=\u22122\u22c5(3\u22120)=\u22126\\int_0^3 -2 \\, dx = -2 \\cdot (3 – 0) = -6\u222b03\u200b\u22122dx=\u22122\u22c5(3\u22120)=\u22126<\/p>\n\n\n\n

    Here, the function is constant and negative, so the integral yields a negative value<\/strong>.<\/p>\n\n\n\n

    Clarification:<\/h4>\n\n\n\n

    If you are only interested in the total area<\/strong> regardless of whether it’s above or below the x-axis, you would use the absolute value<\/strong>:\u222bab\u2223f(x)\u2223\u2009dx\\int_a^b |f(x)| \\, dx\u222bab\u200b\u2223f(x)\u2223dx<\/p>\n\n\n\n

    This always gives a positive result, but it’s a different<\/strong> kind of integral with a different meaning.<\/p>\n\n\n\n

    Conclusion:<\/h3>\n\n\n\n

    The value of a definite integral depends on the sign of the function within the interval. Hence, the statement “The value of a definite integral will always be positive” is false<\/strong>.<\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    The value of a definite integral will always be positive. ? True ? False The Correct Answer and Explanation is: Correct Answer: \u274c False Explanation: The value of a definite integral is not always positive. A definite integral represents the net area between the graph of a function and the x-axis over a given interval […]<\/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-230328","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/230328","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=230328"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/230328\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=230328"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=230328"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=230328"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}