{"id":187175,"date":"2025-02-03T05:27:51","date_gmt":"2025-02-03T05:27:51","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=187175"},"modified":"2025-02-03T05:27:53","modified_gmt":"2025-02-03T05:27:53","slug":"the-following-integral-can-be-evaluated-only-by-reversing-the-order-of-integration","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/03\/the-following-integral-can-be-evaluated-only-by-reversing-the-order-of-integration\/","title":{"rendered":"The following integral can be evaluated only by reversing the order of integration"},"content":{"rendered":"\n

The following integral can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. Integrate 0 to 27 Integrate cube root x to 3 (x\/y^7+1) dy dx Choose the correct sketch of the region below. The reversed order of integration is Integrate Integrate (x\/y^7 + 1) dx dy. (Simplify your answers.) The value of the integral is . (Type an exact answer.)<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

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

To evaluate the integral<\/p>\n\n\n\n

[
\\int_{0}^{27} \\int_{\\sqrt[3]{x}}^{3} \\left( \\frac{x}{y^7} + 1 \\right) dy \\, dx,
]<\/p>\n\n\n\n

we begin by sketching the region of integration in the xy-plane. The outer integral’s limits, ( x ) from 0 to 27, indicate that the region spans horizontally from ( x = 0 ) to ( x = 27 ). The inner integral’s limits, ( y ) from ( \\sqrt[3]{x} ) to 3, suggest that for a fixed ( x ), ( y ) ranges from the curve ( y = \\sqrt[3]{x} ) up to the horizontal line ( y = 3 ).<\/p>\n\n\n\n

To visualize this, plot the curve ( y = \\sqrt[3]{x} ), which passes through points like (0,0), (1,1), (8,2), and (27,3). The region of integration is bounded below by this curve, above by the line ( y = 3 ), on the left by ( x = 0 ), and on the right by ( x = 27 ).<\/p>\n\n\n\n

Next, we reverse the order of integration. To do this, express the region in terms of ( y ) first and then ( x ). Observing the region, ( y ) ranges from 0 to 3. For a fixed ( y ), ( x ) ranges from ( x = 0 ) to ( x = y^3 ) (since ( x = y^3 ) is the inverse of ( y = \\sqrt[3]{x} )). Thus, the reversed order of integration is:<\/p>\n\n\n\n

[
\\int_{0}^{3} \\int_{0}^{y^3} \\left( \\frac{x}{y^7} + 1 \\right) dx \\, dy.
]<\/p>\n\n\n\n

Now, evaluate the inner integral with respect to ( x ):<\/p>\n\n\n\n

[
\\int_{0}^{y^3} \\left( \\frac{x}{y^7} + 1 \\right) dx.
]<\/p>\n\n\n\n

This can be split into two integrals:<\/p>\n\n\n\n

[
\\int_{0}^{y^3} \\frac{x}{y^7} \\, dx + \\int_{0}^{y^3} 1 \\, dx.
]<\/p>\n\n\n\n

Evaluate each separately:<\/p>\n\n\n\n

    \n
  1. [
    \\int_{0}^{y^3} \\frac{x}{y^7} \\, dx = \\frac{1}{y^7} \\int_{0}^{y^3} x \\, dx = \\frac{1}{y^7} \\left[ \\frac{x^2}{2} \\right]_{0}^{y^3} = \\frac{1}{y^7} \\cdot \\frac{(y^3)^2}{2} = \\frac{y^6}{2y^7} = \\frac{1}{2y}.
    ]<\/li>\n\n\n\n
  2. [
    \\int_{0}^{y^3} 1 \\, dx = \\left[ x \\right]_{0}^{y^3} = y^3.
    ]<\/li>\n<\/ol>\n\n\n\n

    Combining these results:<\/p>\n\n\n\n

    [
    \\int_{0}^{y^3} \\left( \\frac{x}{y^7} + 1 \\right) dx = \\frac{1}{2y} + y^3.
    ]<\/p>\n\n\n\n

    Now, integrate this result with respect to ( y ) from 0 to 3:<\/p>\n\n\n\n

    [
    \\int_{0}^{3} \\left( \\frac{1}{2y} + y^3 \\right) dy.
    ]<\/p>\n\n\n\n

    Evaluate each term separately:<\/p>\n\n\n\n

      \n
    1. [
      \\int_{0}^{3} \\frac{1}{2y} \\, dy = \\frac{1}{2} \\int_{0}^{3} \\frac{1}{y} \\, dy = \\frac{1}{2} [\\ln|y|]_{0}^{3}.
      ] As ( y ) approaches 0, ( \\ln|y| ) approaches ( -\\infty ), indicating an improper integral. However, since ( \\frac{1}{y} ) is not integrable at ( y = 0 ), this term diverges.<\/li>\n<\/ol>\n\n\n\n

      Given the divergence in the ( \\frac{1}{2y} ) term, the original integral does not converge to a finite value. Therefore, the value of the integral is divergent.<\/p>\n\n\n\n

      In summary, after reversing the order of integration and attempting to evaluate the integral, we find that the integral diverges due to the ( \\frac{1}{2y} ) term, which is not integrable at ( y = 0 ).<\/p>\n","protected":false},"excerpt":{"rendered":"

      The following integral can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. Integrate 0 to 27 Integrate cube root x to 3 (x\/y^7+1) dy dx Choose the correct sketch of the region below. The reversed order of integration is Integrate […]<\/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-187175","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/187175","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=187175"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/187175\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=187175"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=187175"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=187175"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}