{"id":14186,"date":"2022-08-11T18:27:59","date_gmt":"2022-08-11T10:27:59","guid":{"rendered":"https:\/\/ocw.nycu.edu.tw\/?post_type=course_page&#038;p=14186"},"modified":"2026-06-08T16:48:05","modified_gmt":"2026-06-08T08:48:05","slug":"15-7-triple-integrals-15-8-triple-integrals-in-cylindrical-and-spherical-coordinates","status":"publish","type":"course_page","link":"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-science\/department-of-applied-mathematics\/calculus-ii-english-department-of-applied-mathematics-prof-michael-fuchs\/15-7-triple-integrals-15-8-triple-integrals-in-cylindrical-and-spherical-coordinates","title":{"rendered":"15-7 Triple Integrals, 15-8 Triple Integrals in Cylindrical and Spherical Coordinates"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"14186\" class=\"elementor elementor-14186\">\n\t\t\t\t\t\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-10 elementor-section 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href=\"https:\/\/ocw.nycu.edu.tw\/wp-content\/plugins\/elementor-pro\/assets\/css\/widget-theme-elements.min.css\"><p id=\"breadcrumbs\"><span><span><a href=\"https:\/\/ocw.nycu.edu.tw\/\">Homepage<\/a><\/span><\/span><\/p>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"penci-section penci-disSticky penci-structure-20 elementor-section elementor-top-section elementor-element elementor-element-43d3b5df elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"43d3b5df\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-1 penci-sticky-ct    elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-2e31d00\" data-id=\"2e31d00\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-62afa6e2 elementor-widget elementor-widget-penci-info-box\" data-id=\"62afa6e2\" data-element_type=\"widget\" data-widget_type=\"penci-info-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div id=\"penci_info_box_83234\" class=\"penci-block-vc penci-info-box penci-ibox-float-left penci-view-default penci-shape-circle\">\r\n\t\t\t<div class=\"penci-ibox-inner\">\r\n\t\t\t\t\t\t\t\t<div class=\"penci-ibox-icon penci-ibox-icon--icon penci-icon penci-tibox-text\"><span class=\"penci-ibox-icon-fa\"><\/span><\/div>\t\t\t\t<div class=\"penci-ibox-content-wrap\">\r\n\t\t\t\t\t\t\t\t\t\t<h3 class=\"penci-ibox-title\">15-7 Triple Integrals, 15-8 Triple Integrals in Cylindrical and Spherical Coordinates\n<\/h3>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element 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to Course Homepage<\/span>\r\n\t\t\t\t\t\t\t\t\t<\/span>\r\n\t\t\t<\/a>\r\n\t\t<\/div>\r\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-2 penci-sticky-ct    elementor-column elementor-col-50 elementor-inner-column elementor-element elementor-element-723e657b\" data-id=\"723e657b\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<div class=\"elementor-element elementor-element-1ae1a344 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"1ae1a344\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.13.3 - 28-05-2023 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100%;-webkit-mask-repeat:var(--divider-pattern-repeat);mask-repeat:var(--divider-pattern-repeat);background-color:var(--divider-color);-webkit-mask-image:var(--divider-pattern-url);mask-image:var(--divider-pattern-url)}.elementor-widget-divider--no-spacing{--divider-pattern-size:auto}.elementor-widget-divider--bg-round{--divider-pattern-repeat:round}.rtl .elementor-widget-divider .elementor-divider__text{direction:rtl}.e-con-inner>.elementor-widget-divider,.e-con>.elementor-widget-divider{width:var(--container-widget-width,100%);--flex-grow:var(--container-widget-flex-grow)}<\/style>\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-73059f3c elementor-widget elementor-widget-shortcode\" data-id=\"73059f3c\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\">\n<table id=\"tablepress-22\" class=\"tablepress tablepress-id-22\">\n<thead>\n<tr class=\"row-1 odd\">\n\t<th class=\"column-1\">Week<\/th><th class=\"column-2\">Course Content<\/th><th class=\"column-3\">Course Video<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-hover\">\n<tr class=\"row-2 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Chapter11 Functions and Model<br \/>\n11-1 Sequences<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13883&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-3 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11-2 Series <br \/>\n11-3 The Integral Test and Estimates of Sums<br \/>\n11-4 The Comparison Tests<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13888&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-4 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11-4 The Comparison Tests<br \/>\n11-5 Alternating Series<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13894&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-5 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11-6 Absolute Convergence and the Ratio and Root Tests <br \/>\n11-8 Power Series<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13900&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-6 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11-9 Representations of Functions as Power Series<br \/>\n11-10 Taylor and Maclaurin Series<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13907&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-7 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11-10 Taylor and Maclaurin Series<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13912&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-8 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">11-11 The Binomial Series <br \/>\n11-12 Applications of Taylor Polynomials<br \/>\nChapter12 Vectors and the Geometry of Space<br \/>\n12-1 Three-Dimensional Coordinate Systems <br \/>\n12-2 Vectors<br \/>\n12-3 The Dot Product<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13922&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-9 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">12-4 The Cross Product <br \/>\n12-5 Equations of Lines and Planes<br \/>\n12-6 Cylinders and Quadric Surface<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13930&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-10 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">12-6 Cylinders and Quadric Surface <br \/>\n12-7 Cylindrical and Spherical Coordinates  <br \/>\nChapter13 Vector Functions<br \/>\n13-1 Vector Functions and Space Curves<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13940&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-11 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">13-1 Vector Functions and Space Curves<br \/>\n13-2 Derivatives and Integrals of Vector Functions<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13951&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-12 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">13-2 Derivatives and Integrals of Vector Functions<br \/>\n13-3 Arc Length and Curvature <br \/>\n13-4 Motion in Space:Velocity and Acceleration<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13958&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-13 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Chapter14 Partial Derivatives<br \/>\n14-1 Functions of Several Variables <br \/>\n14-2 Limits and Continuity<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13968&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-14 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">14-3 Partial Derivatives<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13974&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-15 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">14-4 Tangent Lanes and Linear Approximations<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14219&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-16 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">14-5 The Chain Rule <br \/>\n14-6 Directional Derivatives and the Gradient Vector<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13980&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-17 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">14-6 Directional Derivatives and the Gradient Vector  <br \/>\n14-7 Maximum and Minimum Values<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=13989&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-18 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">14-7 Maximum and Minimum Values<br \/>\n14-8 Lagrange Multipliers<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14143&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-19 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">14-8 Lagrange Multipliers<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14149&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-20 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">Chapter15 Multiple Integrals<br \/>\n15-1 Double Integrals over Rectangles <br \/>\n15-2 Iterated Integrals<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14155&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-21 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">15-3 Double Integrals over General Regions <br \/>\n15-4 Double Integrals in Polar Coordinates<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14161&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-22 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">15-6 Surface Area <br \/>\n15-7 Triple Integrals<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14167&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-23 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">15-7 Triple Integrals<br \/>\n15-8 Triple Integrals in Cylindrical and Spherical Coordinates<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14186&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-24 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">15-9 Change of Variables in Multiple Integrals <br \/>\nChapter16 Vector Calculus<br \/>\n16-1 Vector Fields<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14195&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-25 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">16-2 Line Integrals <br \/>\n16-3 The Fundanental Theorem for Line Integrals<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14201&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-26 even\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">16-3 The Fundanental Theorem for Line Integrals  <br \/>\n16-5 Curl and Divergence<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14207&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<tr class=\"row-27 odd\">\n\t<td class=\"column-1\"><\/td><td class=\"column-2\">16-6 Parametric Surface and Their Areas <br \/>\n16-7 Surface Area<br \/>\n16-8 Stokes Theorem<br \/>\n16-9 The Divergence Theorem<\/td><td class=\"column-3\"><a href=\"\/?post_type=course_page&amp;p=14213&amp;lang=en\" rel=\"noopener\" target=\"_blank\">Watch Online<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-22 from cache --><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"penci-ercol-50 penci-ercol-order-3 penci-sticky-ct    elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-48a0b44\" data-id=\"48a0b44\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-62a7c21 elementor-widget-divider--view-line_text elementor-widget-divider--element-align-center elementor-widget elementor-widget-divider\" data-id=\"62a7c21\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span 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class=\"elementor-item\" tabindex=\"-1\">College of Biomedical Science and Engineering<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-132738\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/college-of-life-sciences-en\" class=\"elementor-item\" tabindex=\"-1\">College of Life Sciences<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-141797\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/colege-of-medicine-en\" class=\"elementor-item\" tabindex=\"-1\">College of Medicine<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-118072\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/course-of-chinese-center\" class=\"elementor-item\" tabindex=\"-1\">Course of Chinese Center<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-120431\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/liberal-arts-college\" class=\"elementor-item\" tabindex=\"-1\">Forum Of Liberal Arts<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-123746\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/speech-of-general-education-center\" class=\"elementor-item\" tabindex=\"-1\">Contemporary Civilization<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-128240\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/reading-of-classic-masterpiece\" class=\"elementor-item\" tabindex=\"-1\">Reading of Classic Masterpiece<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-course_page ajax-mega-menu menu-item-123745\"><a href=\"https:\/\/ocw.nycu.edu.tw\/?course_page=all-course-2\/course-of-general-education-center-2\" class=\"elementor-item\" tabindex=\"-1\">Course of General Education Center<\/a><\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>15-7 Triple In&hellip;<\/p>\n","protected":false},"featured_media":0,"parent":13676,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","format":"standard","meta":{"footnotes":""},"categories":[894],"tags":[],"class_list":["post-14186","course_page","type-course_page","status-publish","format-standard","hentry","category-calculus-ii-english-96-en"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.8 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>15-7 Triple Integrals, 15-8 Triple Integrals in Cylindrical and Spherical Coordinates - NYCU 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