{"id":214478,"date":"2025-05-14T12:30:47","date_gmt":"2025-05-14T12:30:47","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=214478"},"modified":"2025-05-14T12:30:51","modified_gmt":"2025-05-14T12:30:51","slug":"evaluate-the-integral-by-changing-to-cylindrical-coordinates","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/14\/evaluate-the-integral-by-changing-to-cylindrical-coordinates\/","title":{"rendered":"Evaluate the integral by changing to cylindrical coordinates."},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Evaluate the integral by changing to cylindrical coordinates.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Integral<sup>2<\/sup><sub>-2<\/sub>&nbsp;integral<sup>squareroot<\/sup>&nbsp;4 &#8211; y<sup>2<\/sup><sub>-squareroot<\/sub>&nbsp;4 &#8211; y<sup>2<\/sup>&nbsp;integral<sup>2<\/sup><sub>squareroot<\/sub>&nbsp;x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;xz dz dx dy<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">integral<sup>3<\/sup><sub>-3<\/sub>&nbsp;integral<sup>squareroot<\/sup>&nbsp;9 &#8211; x<sup>2<\/sup><sub>0<\/sub>&nbsp;integral<sup>9<\/sup>-x<sup>2<\/sup>-y<sup>2<\/sup><sub>0<\/sub>&nbsp;squareroot x<sup>2<\/sup>&nbsp;+ y<sup>2<\/sup>&nbsp;dz dy dx<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To evaluate the given integrals by changing to cylindrical coordinates, we proceed as follows:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Integral 1: \u222b\u221222\u222b\u22124\u2212y24\u2212y2\u222bx2+y24xz\u2009dz\u2009dx\u2009dy\\int_{-2}^{2} \\int_{-\\sqrt{4-y^2}}^{\\sqrt{4-y^2}} \\int_{x^2+y^2}^{4} xz \\, dz \\, dx \\, dy(<a href=\"https:\/\/brainly.com\/question\/44121966?utm_source=chatgpt.com\" target=\"_blank\" rel=\"noopener\">Brainly<\/a>)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 1: Convert to cylindrical coordinates<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>In cylindrical coordinates, x=rcos\u2061\u03b8x = r \\cos\\theta, y=rsin\u2061\u03b8y = r \\sin\\theta, and z=zz = z. The differential element dx\u2009dydx \\, dy transforms to r\u2009dr\u2009d\u03b8r \\, dr \\, d\\theta.(<a href=\"https:\/\/brainly.com\/question\/45042041?utm_source=chatgpt.com\" target=\"_blank\" rel=\"noopener\">Brainly<\/a>)<\/li>\n\n\n\n<li>The integrand xzxz becomes rcos\u2061\u03b8\u22c5zr \\cos\\theta \\cdot z.<\/li>\n\n\n\n<li>The limits for rr range from 0 to 2 (since x2+y2\u22644x^2 + y^2 \\leq 4), and \u03b8\\theta ranges from 0 to 2\u03c02\\pi.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 2: Set up the integral<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The integral becomes:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 3: Evaluate the integral<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First, integrate with respect to zz:<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Now, the integral becomes:<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Next, integrate with respect to rr:<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Finally, integrate with respect to \u03b8\\theta:(<a href=\"https:\/\/math.stackexchange.com\/questions\/1561191\/convert-int2-2-int-sqrt4-x2-sqrt4-x2-int4-x2y2-x?utm_source=chatgpt.com\" target=\"_blank\" rel=\"noopener\">Mathematics Stack Exchange<\/a>)<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Since the integral of cos\u2061\u03b8\\cos\\theta over a full period is zero, the entire integral evaluates to 0.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Integral 2: \u222b\u221233\u222b09\u2212x2\u222b09\u2212x2\u2212y2x2+y2\u2009dz\u2009dy\u2009dx\\int_{-3}^{3} \\int_{0}^{\\sqrt{9-x^2}} \\int_{0}^{\\sqrt{9-x^2-y^2}} \\sqrt{x^2 + y^2} \\, dz \\, dy \\, dx(<a href=\"https:\/\/math.stackexchange.com\/questions\/3880003\/evaluating-int-4-4-int-0-sqrt16-x2-int-0-16-x2-y2-sqrtx2?utm_source=chatgpt.com\" target=\"_blank\" rel=\"noopener\">Mathematics Stack Exchange<\/a>)<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 1: Convert to cylindrical coordinates<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>In cylindrical coordinates, x=rcos\u2061\u03b8x = r \\cos\\theta, y=rsin\u2061\u03b8y = r \\sin\\theta, and z=zz = z. The differential element dx\u2009dydx \\, dy transforms to r\u2009dr\u2009d\u03b8r \\, dr \\, d\\theta.(<a href=\"https:\/\/brainly.com\/question\/45042041?utm_source=chatgpt.com\" target=\"_blank\" rel=\"noopener\">Brainly<\/a>)<\/li>\n\n\n\n<li>The integrand x2+y2\\sqrt{x^2 + y^2} becomes rr.<\/li>\n\n\n\n<li>The limits for rr range from 0 to 3 (since x2+y2\u22649x^2 + y^2 \\leq 9), and \u03b8\\theta ranges from 0 to 2\u03c02\\pi.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 2: Set up the integral<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The integral becomes:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Step 3: Evaluate the integral<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First, integrate with respect to zz:<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Now, the integral becomes:<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Next, integrate with respect to rr:<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">This integral requires a substitution. Let u=9\u2212r2u = 9 &#8211; r^2, so du=\u22122r\u2009drdu = -2r \\, dr. The limits change accordingly: when r=0r = 0, u=9u = 9; and when r=3r = 3, u=0u = 0. The integral becomes:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Finally, integrate with respect to \u03b8\\theta:<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Multiplying all parts together:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Conclusion<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The value of the first integral is 0.<\/li>\n\n\n\n<li>The value of the second integral is 18\u03c018\\pi.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">These results demonstrate the application of cylindrical coordinates to simplify and evaluate double and triple integrals over circular and cylindrical regions.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Evaluate the integral by changing to cylindrical coordinates. Integral2-2&nbsp;integralsquareroot&nbsp;4 &#8211; y2-squareroot&nbsp;4 &#8211; y2&nbsp;integral2squareroot&nbsp;x2&nbsp;+ y2&nbsp;xz dz dx dy integral3-3&nbsp;integralsquareroot&nbsp;9 &#8211; x20&nbsp;integral9-x2-y20&nbsp;squareroot x2&nbsp;+ y2&nbsp;dz dy dx The Correct Answer and Explanation is: To evaluate the given integrals by changing to cylindrical coordinates, we proceed as follows: Integral 1: \u222b\u221222\u222b\u22124\u2212y24\u2212y2\u222bx2+y24xz\u2009dz\u2009dx\u2009dy\\int_{-2}^{2} \\int_{-\\sqrt{4-y^2}}^{\\sqrt{4-y^2}} \\int_{x^2+y^2}^{4} xz \\, dz \\, dx \\, [&hellip;]<\/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-214478","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/214478","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=214478"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/214478\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=214478"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=214478"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=214478"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}