Evaluate the triple integral:<\/p>\n\n\n\n
\u222b \u222b \u222b E xy dV, where E is bounded by the parabolic cylinders y =x2 and x = y2 and the planes z = 0 and z = x+ y.<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n The given triple integral is:<\/p>\n\n\n\n [ where ( E ) is the region bounded by the parabolic cylinders ( y = x^2 ), ( x = y^2 ), and the planes ( z = 0 ), and ( z = x + y ). We need to evaluate the integral.<\/p>\n\n\n\n We first analyze the boundaries of the region described by the parabolic cylinders.<\/p>\n\n\n\n From the analysis above, the bounds for the triple integral are:<\/p>\n\n\n\n [ We evaluate the integral step by step:<\/p>\n\n\n\n [ [ [ Summing these integrals gives:<\/p>\n\n\n\n [ Now we integrate each term with respect to ( x ) from 0 to 1:<\/p>\n\n\n\n [ The integrals are:<\/p>\n\n\n\n [ [ [ [ Summing these results gives the value of the triple integral:<\/p>\n\n\n\n [ Thus, the value of the triple integral is:<\/p>\n\n\n\n [ Evaluate the triple integral: \u222b \u222b \u222b E xy dV, where E is bounded by the parabolic cylinders y =x2 and x = y2 and the planes z = 0 and z = x+ y. The Correct Answer and Explanation is : The given triple integral is: [\\int \\int \\int_E xy \\, dV] where ( […]<\/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-190096","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190096","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=190096"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/190096\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=190096"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=190096"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=190096"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\\int \\int \\int_E xy \\, dV
]<\/p>\n\n\n\nStep 1: Understand the boundaries of the region<\/h3>\n\n\n\n
\n
\n
\n
Step 2: Set up the triple integral<\/h3>\n\n\n\n
\\int_0^1 \\int_{x^2}^{\\sqrt{x}} \\int_0^{x + y} xy \\, dz \\, dy \\, dx
]<\/p>\n\n\n\nStep 3: Evaluate the integral<\/h3>\n\n\n\n
\n
\\int_0^{x+y} xy \\, dz = xy(x + y)
]<\/p>\n\n\n\n\n
\\int_{x^2}^{\\sqrt{x}} xy(x + y) \\, dy
]
We expand the integrand:
[
xy(x + y) = x^2 y + x y^2
]
Now integrate term by term:
[
\\int_{x^2}^{\\sqrt{x}} x^2 y \\, dy = x^2 \\left[ \\frac{y^2}{2} \\right]_{x^2}^{\\sqrt{x}} = x^2 \\left( \\frac{x}{2} – \\frac{x^4}{2} \\right) = \\frac{x^3}{2} – \\frac{x^6}{2}
]<\/p>\n\n\n\n
\\int_{x^2}^{\\sqrt{x}} x y^2 \\, dy = x \\left[ \\frac{y^3}{3} \\right]_{x^2}^{\\sqrt{x}} = x \\left( \\frac{x^{3\/2}}{3} – \\frac{x^6}{3} \\right) = \\frac{x^{5\/2}}{3} – \\frac{x^7}{3}
]<\/p>\n\n\n\n
\\frac{x^3}{2} – \\frac{x^6}{2} + \\frac{x^{5\/2}}{3} – \\frac{x^7}{3}
]<\/p>\n\n\n\n\n
\\int_0^1 \\left( \\frac{x^3}{2} – \\frac{x^6}{2} + \\frac{x^{5\/2}}{3} – \\frac{x^7}{3} \\right) dx
]<\/p>\n\n\n\n
\\int_0^1 \\frac{x^3}{2} \\, dx = \\frac{1}{2} \\cdot \\frac{x^4}{4} \\Big|_0^1 = \\frac{1}{8}
]<\/p>\n\n\n\n
\\int_0^1 \\frac{x^6}{2} \\, dx = \\frac{1}{2} \\cdot \\frac{x^7}{7} \\Big|_0^1 = \\frac{1}{14}
]<\/p>\n\n\n\n
\\int_0^1 \\frac{x^{5\/2}}{3} \\, dx = \\frac{1}{3} \\cdot \\frac{x^{7\/2}}{7\/2} \\Big|_0^1 = \\frac{2}{21}
]<\/p>\n\n\n\n
\\int_0^1 \\frac{x^7}{3} \\, dx = \\frac{1}{3} \\cdot \\frac{x^8}{8} \\Big|_0^1 = \\frac{1}{24}
]<\/p>\n\n\n\nStep 4: Sum up the results<\/h3>\n\n\n\n
\\frac{1}{8} – \\frac{1}{14} + \\frac{2}{21} – \\frac{1}{24} = \\frac{147}{168} – \\frac{12}{168} + \\frac{16}{168} – \\frac{7}{168} = \\frac{144}{168} = \\frac{6}{7}
]<\/p>\n\n\n\n
\\boxed{\\frac{6}{7}}
]<\/p>\n","protected":false},"excerpt":{"rendered":"