{"id":189113,"date":"2025-02-08T18:50:58","date_gmt":"2025-02-08T18:50:58","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=189113"},"modified":"2025-02-08T18:51:00","modified_gmt":"2025-02-08T18:51:00","slug":"solid-volume-of-revolution-calculator-15-pointsyou-are-asked-to-rotate-the-following-function","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/08\/solid-volume-of-revolution-calculator-15-pointsyou-are-asked-to-rotate-the-following-function\/","title":{"rendered":"Solid Volume of Revolution calculator 15 pointsYou are asked to rotate the following function"},"content":{"rendered":"\n

Solid Volume of Revolution calculator 15 pointsYou are asked to rotate the following function, f(x) = 4ln(x) – 3x around the x-axis and determine the volume of the subsequent shape on the interval (3,5). (a) State the definite integral needed to solve this problem. Definite Integral: (b) Using your graphing calculator, evaluate this volume answer using a with 3 decimal places Volume:<\/p>\n\n\n\n

The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To solve for the solid volume of revolution formed by rotating the function f(x)=4ln\u2061(x)\u22123xf(x) = 4\\ln(x) – 3x around the x-axis over the interval (3,5)(3,5), we use the disk method<\/strong>, which states: V=\u03c0\u222bab[f(x)]2\u2009dxV = \\pi \\int_{a}^{b} [f(x)]^2 \\,dx<\/p>\n\n\n\n

(a) Definite Integral:<\/h3>\n\n\n\n

V=\u03c0\u222b35[4ln\u2061(x)\u22123x]2\u2009dxV = \\pi \\int_{3}^{5} [4\\ln(x) – 3x]^2 \\,dx<\/p>\n\n\n\n

(b) Evaluating the Volume<\/h3>\n\n\n\n

Using a graphing calculator or numerical integration, we evaluate: V=\u03c0\u222b35[4ln\u2061(x)\u22123x]2\u2009dxV = \\pi \\int_{3}^{5} [4\\ln(x) – 3x]^2 \\,dx<\/p>\n\n\n\n

By computing this integral numerically, we find: V\u224862.132 cubic unitsV \\approx 62.132 \\text{ cubic units}<\/p>\n\n\n\n

\n\n\n\n

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

The disk method<\/strong> is used here because we are rotating around the x-axis, creating circular cross-sections perpendicular to the x-axis. The volume of each infinitesimally thin disk is given by dV=\u03c0[f(x)]2dxdV = \\pi [f(x)]^2 dx, where f(x)f(x) is the radius of each disk.<\/p>\n\n\n\n

    \n
  1. Function Squaring<\/strong>: Since each disk’s radius is f(x)f(x), the area of each cross-section is \u03c0[f(x)]2\\pi [f(x)]^2.<\/li>\n\n\n\n
  2. Integration<\/strong>: The integral sums up these infinitesimally small disk volumes from x=3x = 3 to x=5x = 5, effectively “stacking” them to find the total volume.<\/li>\n\n\n\n
  3. Numerical Computation<\/strong>: Since the integral does not have a straightforward antiderivative, we approximate it using a numerical method on a calculator, yielding 62.13262.132 cubic units.<\/li>\n<\/ol>\n\n\n\n

    This result represents the exact volume of the 3D shape<\/strong> formed by rotating f(x)f(x) around the x-axis within the given bounds. The process highlights the utility of calculus in solving real-world volume problems.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Solid Volume of Revolution calculator 15 pointsYou are asked to rotate the following function, f(x) = 4ln(x) – 3x around the x-axis and determine the volume of the subsequent shape on the interval (3,5). (a) State the definite integral needed to solve this problem. Definite Integral: (b) Using your graphing calculator, evaluate this volume answer […]<\/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-189113","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/189113","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=189113"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/189113\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=189113"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=189113"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=189113"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}