{"id":1267,"date":"2025-10-15T15:57:07","date_gmt":"2025-10-15T15:57:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=1267"},"modified":"2025-10-31T16:21:14","modified_gmt":"2025-10-31T16:21:14","slug":"alternative-coordinate-systems-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/alternative-coordinate-systems-learn-it-3\/","title":{"raw":"Triple Integrals in Alternative Coordinate Systems: Learn It 3","rendered":"Triple Integrals in Alternative Coordinate Systems: Learn It 3"},"content":{"raw":"<h2 data-type=\"title\">Review of Spherical Coordinates<\/h2>\r\nIn three-dimensional space [latex]{\\mathbb{R}^{3}}[\/latex] in the spherical coordinate system, we specify a point [latex]P[\/latex] by its distance [latex]{\\rho}[\/latex] from the origin, the polar angle [latex]{\\theta}[\/latex] from the positive [latex]x[\/latex]-axis (same as in the cylindrical coordinate system), and the angle [latex]{\\varphi}[\/latex] from the positive [latex]z[\/latex]-axis and the line [latex]OP[\/latex] (Figure 1). Note that [latex]{\\rho} \\ {\\geq} \\ {0}[\/latex] and [latex]{0} \\ {\\leq} \\ {\\varphi} \\ {\\leq} \\ {\\pi}[\/latex].\r\n\r\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>When to Use Spherical Coordinates? <\/strong>Spherical coordinates are useful for triple integrals over regions that are symmetric with respect to the origin, such as spheres, cones, and other solids centered at the origin.<\/section>\r\n\r\n[caption id=\"attachment_1396\" align=\"aligncenter\" width=\"394\"]<img class=\"size-full wp-image-1396\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/26173418\/5-5-6.jpeg\" alt=\"A depiction of the spherical coordinate system: a point (x, y, z) is shown, which is equal to (rho, theta, phi) in spherical coordinates. Rho serves as the spherical radius, theta serves as the angle from the x axis in the xy plane, and phi serves as the angle from the z axis.\" width=\"394\" height=\"371\" \/> Figure 1.\u00a0The spherical coordinate system locates points with two angles and a distance from the origin.[\/caption]\r\n<p id=\"fs-id1167794181221\">Recall the relationships that connect rectangular coordinates with spherical coordinates.<\/p>\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\"><strong>From spherical to rectangular coordinates:<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x = {\\rho} \\sin \\varphi \\cos \\theta[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y = {\\rho} \\sin \\varphi \\sin \\theta[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z = {\\rho} \\cos \\varphi[\/latex]<\/li>\r\n<\/ul>\r\n<strong>From rectangular to spherical coordinates:<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">[latex]{\\rho}^{2} = {x^2} + {y^2} + {z^2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\tan\\theta = \\frac{y}{x}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\varphi = \\arccos\\left ( \\frac{z}{\\sqrt{{x^2} + {y^2} + {z^2}}} \\right )[\/latex]<\/li>\r\n<\/ul>\r\n<strong>From spherical to cylindrical coordinates:<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r = {\\rho} \\sin \\varphi[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\theta = \\theta[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z = {\\rho} \\cos \\varphi[\/latex]<\/li>\r\n<\/ul>\r\n<strong>From cylindrical to spherical coordinates:<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\rho = \\sqrt{{r^2}+{z^2}}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\theta = \\theta[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\varphi = \\arccos\\left ( \\frac{z}{\\sqrt{{r^2}+{z^2}}} \\right )[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><span style=\"font-size: 1rem; text-align: initial;\">The following figure shows a few solid regions that are convenient to express in spherical coordinates.<\/span>\r\n<div id=\"fs-id1167794050870\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\r\n<div class=\"MathJax_Display\"><\/div>\r\n<\/div>\r\n[caption id=\"attachment_1397\" align=\"aligncenter\" width=\"923\"]<img class=\"size-full wp-image-1397\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/26173500\/5-5-7.jpeg\" alt=\"This figure consists of four figures. In the first, a sphere is shown with the note Sphere rho = c (constant). In the second, a half plane is drawn from the z axis with the note Half plane theta = c (constant). In the last two figures, a half cone is drawn in each with the note Half cone phi = c (constant). In the first of these, the cone opens up and it is marked 0 &lt; c &lt; pi\/2. In the second of these, the cone opens down and it is marked pi\/2 &lt; c &lt; pi.\" width=\"923\" height=\"290\" \/> Figure 2.\u00a0Spherical coordinates are especially convenient for working with solids bounded by these types of surfaces. (The letter [latex]c[\/latex] indicates a constant.)[\/caption]<section class=\"textbox proTip\" aria-label=\"Pro Tip\">For a more detailed review of cylindrical and spherical coordinates, including how to visualize these coordinate systems and work with them in three dimensions, refer to the section on Alternative Coordinate Systems and Surfaces.<\/section>","rendered":"<h2 data-type=\"title\">Review of Spherical Coordinates<\/h2>\n<p>In three-dimensional space [latex]{\\mathbb{R}^{3}}[\/latex] in the spherical coordinate system, we specify a point [latex]P[\/latex] by its distance [latex]{\\rho}[\/latex] from the origin, the polar angle [latex]{\\theta}[\/latex] from the positive [latex]x[\/latex]-axis (same as in the cylindrical coordinate system), and the angle [latex]{\\varphi}[\/latex] from the positive [latex]z[\/latex]-axis and the line [latex]OP[\/latex] (Figure 1). Note that [latex]{\\rho} \\ {\\geq} \\ {0}[\/latex] and [latex]{0} \\ {\\leq} \\ {\\varphi} \\ {\\leq} \\ {\\pi}[\/latex].<\/p>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>When to Use Spherical Coordinates? <\/strong>Spherical coordinates are useful for triple integrals over regions that are symmetric with respect to the origin, such as spheres, cones, and other solids centered at the origin.<\/section>\n<figure id=\"attachment_1396\" aria-describedby=\"caption-attachment-1396\" style=\"width: 394px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1396\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/26173418\/5-5-6.jpeg\" alt=\"A depiction of the spherical coordinate system: a point (x, y, z) is shown, which is equal to (rho, theta, phi) in spherical coordinates. Rho serves as the spherical radius, theta serves as the angle from the x axis in the xy plane, and phi serves as the angle from the z axis.\" width=\"394\" height=\"371\" \/><figcaption id=\"caption-attachment-1396\" class=\"wp-caption-text\">Figure 1.\u00a0The spherical coordinate system locates points with two angles and a distance from the origin.<\/figcaption><\/figure>\n<p id=\"fs-id1167794181221\">Recall the relationships that connect rectangular coordinates with spherical coordinates.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\"><strong>From spherical to rectangular coordinates:<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]x = {\\rho} \\sin \\varphi \\cos \\theta[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y = {\\rho} \\sin \\varphi \\sin \\theta[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z = {\\rho} \\cos \\varphi[\/latex]<\/li>\n<\/ul>\n<p><strong>From rectangular to spherical coordinates:<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]{\\rho}^{2} = {x^2} + {y^2} + {z^2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\tan\\theta = \\frac{y}{x}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\varphi = \\arccos\\left ( \\frac{z}{\\sqrt{{x^2} + {y^2} + {z^2}}} \\right )[\/latex]<\/li>\n<\/ul>\n<p><strong>From spherical to cylindrical coordinates:<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]r = {\\rho} \\sin \\varphi[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\theta = \\theta[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z = {\\rho} \\cos \\varphi[\/latex]<\/li>\n<\/ul>\n<p><strong>From cylindrical to spherical coordinates:<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]\\rho = \\sqrt{{r^2}+{z^2}}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\theta = \\theta[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\varphi = \\arccos\\left ( \\frac{z}{\\sqrt{{r^2}+{z^2}}} \\right )[\/latex]<\/li>\n<\/ul>\n<\/section>\n<p><span style=\"font-size: 1rem; text-align: initial;\">The following figure shows a few solid regions that are convenient to express in spherical coordinates.<\/span><\/p>\n<div id=\"fs-id1167794050870\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\n<div class=\"MathJax_Display\"><\/div>\n<\/div>\n<figure id=\"attachment_1397\" aria-describedby=\"caption-attachment-1397\" style=\"width: 923px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1397\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/26173500\/5-5-7.jpeg\" alt=\"This figure consists of four figures. In the first, a sphere is shown with the note Sphere rho = c (constant). In the second, a half plane is drawn from the z axis with the note Half plane theta = c (constant). In the last two figures, a half cone is drawn in each with the note Half cone phi = c (constant). In the first of these, the cone opens up and it is marked 0 &lt; c &lt; pi\/2. In the second of these, the cone opens down and it is marked pi\/2 &lt; c &lt; pi.\" width=\"923\" height=\"290\" \/><figcaption id=\"caption-attachment-1397\" class=\"wp-caption-text\">Figure 2.\u00a0Spherical coordinates are especially convenient for working with solids bounded by these types of surfaces. (The letter [latex]c[\/latex] indicates a constant.)<\/figcaption><\/figure>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">For a more detailed review of cylindrical and spherical coordinates, including how to visualize these coordinate systems and work with them in three dimensions, refer to the section on Alternative Coordinate Systems and Surfaces.<\/section>\n","protected":false},"author":15,"menu_order":13,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1267","chapter","type-chapter","status-publish","hentry"],"part":434,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1267","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1267\/revisions"}],"predecessor-version":[{"id":1936,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1267\/revisions\/1936"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/434"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1267\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=1267"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=1267"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=1267"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=1267"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}