{"id":20233,"date":"2024-04-15T05:50:11","date_gmt":"2024-04-15T05:50:11","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20233"},"modified":"2024-04-15T05:50:11","modified_gmt":"2024-04-15T05:50:11","slug":"the-volume-of-the-solid-surrounded-by-the-surface-left-fracrmxrma-rightfrac23-left-fracrmyrmb-rightfrac23-left-f","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-volume-of-the-solid-surrounded-by-the-surface-left-fracrmxrma-rightfrac23-left-fracrmyrmb-rightfrac23-left-f\/","title":{"rendered":"The volume of the solid surrounded by the surface \\[{\\left( {\\frac{{\\rm{x}}}{{\\rm{a}}}} \\right)^{\\frac{2}{3}}} + {\\left( {\\frac{{\\rm{y}}}{{\\rm{b}}}} \\right)^{\\frac{2}{3}}} + {\\left( {\\frac{{\\rm{z}}}{{\\rm{c}}}} \\right)^{\\frac{2}{3}}} = 1\\] is A. \\[\\frac{{4\\pi {\\rm{abc}}}}{{35}}\\] B. \\[\\frac{{{\\rm{abc}}}}{{35}}\\] C. \\[\\frac{{2\\pi {\\rm{abc}}}}{{35}}\\] D. \\[\\frac{{\\pi {\\rm{abc}}}}{{35}}\\]"},"content":{"rendered":"<p>[amp_mcq option1=&#8221;\\[\\frac{{4\\pi {\\rm{abc}}}}{{35}}\\]&#8221; option2=&#8221;\\[\\frac{{{\\rm{abc}}}}{{35}}\\]&#8221; option3=&#8221;\\[\\frac{{2\\pi {\\rm{abc}}}}{{35}}\\]&#8221; option4=&#8221;\\[\\frac{{\\pi {\\rm{abc}}}}{{35}}\\]&#8221; correct=&#8221;option1&#8243;]<!--more--><\/p>\n<p>The correct answer is $\\boxed{\\frac{{\\pi {\\rm{abc}}}}{{35}}}$.<\/p>\n<p>The surface $x^2\/a^2 + y^2\/b^2 + z^2\/c^2 = 1$ is an ellipsoid. The volume of an ellipsoid is given by the formula<\/p>\n<p>$$V = \\frac{4}{3} \\pi abc \\int_0^1 \\int_0^1 \\int_0^1 \\sqrt{x^2\/a^2 + y^2\/b^2 + z^2\/c^2} dx dy dz.$$<\/p>\n<p>In this case, we have $x^2\/a^2 + y^2\/b^2 + z^2\/c^2 = 1$, so we can simplify the integral to<\/p>\n<p>$$V = \\frac{4}{3} \\pi abc \\int_0^1 \\int_0^1 \\int_0^1 \\sqrt{1} dx dy dz = \\frac{4}{3} \\pi abc.$$<\/p>\n<p>The other options are incorrect because they do not take into account the fact that the surface is an ellipsoid.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>[amp_mcq option1=&#8221;\\[\\frac{{4\\pi {\\rm{abc}}}}{{35}}\\]&#8221; option2=&#8221;\\[\\frac{{{\\rm{abc}}}}{{35}}\\]&#8221; option3=&#8221;\\[\\frac{{2\\pi {\\rm{abc}}}}{{35}}\\]&#8221; option4=&#8221;\\[\\frac{{\\pi {\\rm{abc}}}}{{35}}\\]&#8221; correct=&#8221;option1&#8243;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[690],"tags":[],"class_list":["post-20233","post","type-post","status-publish","format-standard","hentry","category-calculus","no-featured-image-padding"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>The volume of the solid surrounded by the surface \\[{\\left( {\\frac{{\\rm{x}}}{{\\rm{a}}}} \\right)^{\\frac{2}{3}}} + {\\left( {\\frac{{\\rm{y}}}{{\\rm{b}}}} \\right)^{\\frac{2}{3}}} + {\\left( {\\frac{{\\rm{z}}}{{\\rm{c}}}} \\right)^{\\frac{2}{3}}} = 1\\] is A. \\[\\frac{{4\\pi {\\rm{abc}}}}{{35}}\\] B. \\[\\frac{{{\\rm{abc}}}}{{35}}\\] C. \\[\\frac{{2\\pi {\\rm{abc}}}}{{35}}\\] D. \\[\\frac{{\\pi {\\rm{abc}}}}{{35}}\\]<\/title>\n<meta name=\"robots\" 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surface \\[{\\left( {\\frac{{\\rm{x}}}{{\\rm{a}}}} \\right)^{\\frac{2}{3}}} + {\\left( {\\frac{{\\rm{y}}}{{\\rm{b}}}} \\right)^{\\frac{2}{3}}} + {\\left( {\\frac{{\\rm{z}}}{{\\rm{c}}}} \\right)^{\\frac{2}{3}}} = 1\\] is A. \\[\\frac{{4\\pi {\\rm{abc}}}}{{35}}\\] B. \\[\\frac{{{\\rm{abc}}}}{{35}}\\] C. \\[\\frac{{2\\pi {\\rm{abc}}}}{{35}}\\] D. \\[\\frac{{\\pi {\\rm{abc}}}}{{35}}\\]","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/the-volume-of-the-solid-surrounded-by-the-surface-left-fracrmxrma-rightfrac23-left-fracrmyrmb-rightfrac23-left-f\/","og_locale":"en_US","og_type":"article","og_title":"The volume of the solid surrounded by the surface \\[{\\left( {\\frac{{\\rm{x}}}{{\\rm{a}}}} \\right)^{\\frac{2}{3}}} + {\\left( {\\frac{{\\rm{y}}}{{\\rm{b}}}} \\right)^{\\frac{2}{3}}} + {\\left( {\\frac{{\\rm{z}}}{{\\rm{c}}}} \\right)^{\\frac{2}{3}}} = 1\\] is A. \\[\\frac{{4\\pi {\\rm{abc}}}}{{35}}\\] B. \\[\\frac{{{\\rm{abc}}}}{{35}}\\] C. \\[\\frac{{2\\pi {\\rm{abc}}}}{{35}}\\] D. \\[\\frac{{\\pi {\\rm{abc}}}}{{35}}\\]","og_description":"[amp_mcq option1=&#8221;[frac{{4pi {rm{abc}}}}{{35}}]&#8221; option2=&#8221;[frac{{{rm{abc}}}}{{35}}]&#8221; option3=&#8221;[frac{{2pi {rm{abc}}}}{{35}}]&#8221; option4=&#8221;[frac{{pi {rm{abc}}}}{{35}}]&#8221; correct=&#8221;option1&#8243;]","og_url":"https:\/\/exam.pscnotes.com\/mcq\/the-volume-of-the-solid-surrounded-by-the-surface-left-fracrmxrma-rightfrac23-left-fracrmyrmb-rightfrac23-left-f\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T05:50:11+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"1 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