{"id":716,"date":"2025-10-09T15:34:15","date_gmt":"2025-10-09T15:34:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=716"},"modified":"2025-10-31T16:20:57","modified_gmt":"2025-10-31T16:20:57","slug":"alternative-coordinate-systems-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/alternative-coordinate-systems-learn-it-1\/","title":{"raw":"Triple Integrals in Alternative Coordinate Systems: Learn It 1","rendered":"Triple Integrals in Alternative Coordinate Systems: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Evaluate a triple integral by changing to cylindrical coordinates<\/li>\r\n \t<li>Evaluate a triple integral by changing to spherical coordinates<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Triple Integrals in Cylindrical and Spherical Coordinates<\/h2>\r\n<p class=\"whitespace-normal break-words\">We have seen how to convert a double integral in rectangular coordinates into a double integral in polar coordinates in order to deal more conveniently with problems involving circular symmetry. A similar situation occurs with triple integrals, but here we need to distinguish between cylindrical symmetry and spherical symmetry. In this section we convert triple integrals in rectangular coordinates into a triple integral in either cylindrical or spherical coordinates.<\/p>\r\n\r\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">The opera house l'Hemisph\u00e8ric in Valencia, Spain, has four sections with one of the sections being a theater in a five-story-high sphere (ball) under an oval roof as long as a football field. Inside is an IMAX screen that changes the sphere into a planetarium with a sky full of [latex]9000[\/latex] twinkling stars. Using triple integrals in spherical coordinates, we can find the volumes of different geometric shapes like these.\r\n\r\n[caption id=\"attachment_233\" align=\"aligncenter\" width=\"800\"]<img class=\"wp-image-233\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/07\/02173001\/c3-m5.jpeg\" alt=\"A photo of a glass ovoid dome reflecting in the water to make something like a football shape.\" width=\"800\" height=\"539\" \/> Figure 1. The City of Arts and Sciences in Valencia, Spain, has a unique structure along an axis of just two kilometers that was formerly the bed of the River Turia. The l\u2019Hemisf\u00e8ric has an IMAX cinema with three systems of modern digital projections onto a concave screen of 900 square meters. An oval roof over 100 meters long has been made to look like a huge human eye that comes alive and opens up to the world as the \u201cEye of Wisdom.\u201d (credit: modification of work by Javier Yaya Tur, Wikimedia Commons)[\/caption]\r\n\r\n<\/section>\r\n<h3 data-type=\"title\">Review of Cylindrical Coordinates<\/h3>\r\n<p class=\"whitespace-normal break-words\">As we have seen earlier, in two-dimensional space [latex]\\mathbb{R}^2[\/latex], a point with rectangular coordinates [latex](x, y)[\/latex] can be identified with [latex]{({r},{\\theta})}[\/latex] in polar coordinates and vice versa, where [latex]{x} = {r}{\\cos}{\\theta}[\/latex], [latex]{y} = {r}{\\sin}{\\theta}[\/latex], [latex]{r^2} = {x^2}+{y^2}[\/latex], and [latex]{\\tan}{\\theta} = {({\\frac{y}{x}})}[\/latex] are the relationships between the variables.<\/p>\r\n<p class=\"whitespace-normal break-words\">In three-dimensional space [latex]\\mathbb{R}^3[\/latex], a point with rectangular coordinates [latex](x, y, z)[\/latex] can be identified with <strong>cylindrical coordinates<\/strong> [latex]{({r},{\\theta},{z})}[\/latex] and vice versa. We can use these same conversion relationships, adding [latex]z[\/latex] as the vertical distance to the point from the [latex]xy[\/latex]-plane as shown in the following figure.<\/p>\r\n\r\n[caption id=\"attachment_1389\" align=\"aligncenter\" width=\"391\"]<img class=\"size-full wp-image-1389\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/26172826\/5-5-1.jpeg\" alt=\"A box\u00a0[latex]E[\/latex] where the projection\u00a0[latex]D[\/latex] in the\u00a0[latex]xy[\/latex]-plane is of Type I.\" width=\"391\" height=\"391\" \/> Figure 2. Cylindrical coordinates are similar to polar coordinates with a vertical [latex]z[\/latex] coordinate added.[\/caption]<section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p class=\"whitespace-normal break-words\"><strong>From rectangular to cylindrical:<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]{x} = {r}{\\cos}{\\theta}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]{y} = {r}{\\sin}{\\theta}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z = z[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>From cylindrical to rectangular:<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]{r^2} = {x^2} + {y^2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]{\\theta} = {\\tan^{-1}}{({\\frac{y}{x}})}[\/latex] (note: you may need to add [latex]\\pi[\/latex] to reach the correct quadrant)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z = z[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">In the two-dimensional plane with a rectangular coordinate system, when we say [latex]x=k[\/latex] (constant) we mean an unbounded vertical line parallel to the [latex]y[\/latex]-axis and when [latex]y=l[\/latex] (constant) we mean an unbounded horizontal line parallel to the [latex]x[\/latex]-axis. With the polar coordinate system, when we say [latex]r=c[\/latex] (constant), we mean a circle of radius [latex]c[\/latex] units and when [latex]{\\theta} = {\\alpha}[\/latex] (constant) we mean an infinite ray making an angle [latex]{\\alpha}[\/latex] with the positive [latex]x[\/latex]-axis.<\/p>\r\n<p class=\"whitespace-normal break-words\">Similarly, in three-dimensional space with rectangular coordinates [latex](x, y, z)[\/latex], the equations [latex]x=k[\/latex], [latex]y=l[\/latex], and [latex]z=m[\/latex], where [latex]k[\/latex], [latex]l[\/latex], and [latex]m[\/latex] are constants, represent unbounded planes parallel to the [latex]yz[\/latex]-plane, [latex]xz[\/latex]-plane, and [latex]xy[\/latex]-plane, respectively.<\/p>\r\n<p class=\"whitespace-normal break-words\">With cylindrical coordinates [latex]{({r},{\\theta},{z})}[\/latex], the equations [latex]{r} = {c}[\/latex], [latex]{\\theta} = {\\alpha}[\/latex], and [latex]z=m[\/latex], where [latex]{c}[\/latex], [latex]{\\alpha}[\/latex], and [latex]m[\/latex] are constants, represent:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]r = c[\/latex]: an unbounded vertical cylinder with the [latex]z[\/latex]-axis as its radial axis<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\theta = \\alpha[\/latex]: a plane making a constant angle [latex]{\\alpha}[\/latex] with the [latex]xy[\/latex]-plane<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z = m[\/latex]: an unbounded horizontal plane parallel to the [latex]xy[\/latex]-plane<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">This means that the circular cylinder [latex]x^{2}+y^{2}=c^{2}[\/latex] in rectangular coordinates can be represented simply as [latex]r=c[\/latex] in cylindrical coordinates.<\/p>\r\n\r\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>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Evaluate a triple integral by changing to cylindrical coordinates<\/li>\n<li>Evaluate a triple integral by changing to spherical coordinates<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Triple Integrals in Cylindrical and Spherical Coordinates<\/h2>\n<p class=\"whitespace-normal break-words\">We have seen how to convert a double integral in rectangular coordinates into a double integral in polar coordinates in order to deal more conveniently with problems involving circular symmetry. A similar situation occurs with triple integrals, but here we need to distinguish between cylindrical symmetry and spherical symmetry. In this section we convert triple integrals in rectangular coordinates into a triple integral in either cylindrical or spherical coordinates.<\/p>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">The opera house l&#8217;Hemisph\u00e8ric in Valencia, Spain, has four sections with one of the sections being a theater in a five-story-high sphere (ball) under an oval roof as long as a football field. Inside is an IMAX screen that changes the sphere into a planetarium with a sky full of [latex]9000[\/latex] twinkling stars. Using triple integrals in spherical coordinates, we can find the volumes of different geometric shapes like these.<\/p>\n<figure id=\"attachment_233\" aria-describedby=\"caption-attachment-233\" style=\"width: 800px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-233\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/07\/02173001\/c3-m5.jpeg\" alt=\"A photo of a glass ovoid dome reflecting in the water to make something like a football shape.\" width=\"800\" height=\"539\" \/><figcaption id=\"caption-attachment-233\" class=\"wp-caption-text\">Figure 1. The City of Arts and Sciences in Valencia, Spain, has a unique structure along an axis of just two kilometers that was formerly the bed of the River Turia. The l\u2019Hemisf\u00e8ric has an IMAX cinema with three systems of modern digital projections onto a concave screen of 900 square meters. An oval roof over 100 meters long has been made to look like a huge human eye that comes alive and opens up to the world as the \u201cEye of Wisdom.\u201d (credit: modification of work by Javier Yaya Tur, Wikimedia Commons)<\/figcaption><\/figure>\n<\/section>\n<h3 data-type=\"title\">Review of Cylindrical Coordinates<\/h3>\n<p class=\"whitespace-normal break-words\">As we have seen earlier, in two-dimensional space [latex]\\mathbb{R}^2[\/latex], a point with rectangular coordinates [latex](x, y)[\/latex] can be identified with [latex]{({r},{\\theta})}[\/latex] in polar coordinates and vice versa, where [latex]{x} = {r}{\\cos}{\\theta}[\/latex], [latex]{y} = {r}{\\sin}{\\theta}[\/latex], [latex]{r^2} = {x^2}+{y^2}[\/latex], and [latex]{\\tan}{\\theta} = {({\\frac{y}{x}})}[\/latex] are the relationships between the variables.<\/p>\n<p class=\"whitespace-normal break-words\">In three-dimensional space [latex]\\mathbb{R}^3[\/latex], a point with rectangular coordinates [latex](x, y, z)[\/latex] can be identified with <strong>cylindrical coordinates<\/strong> [latex]{({r},{\\theta},{z})}[\/latex] and vice versa. We can use these same conversion relationships, adding [latex]z[\/latex] as the vertical distance to the point from the [latex]xy[\/latex]-plane as shown in the following figure.<\/p>\n<figure id=\"attachment_1389\" aria-describedby=\"caption-attachment-1389\" style=\"width: 391px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1389\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/26172826\/5-5-1.jpeg\" alt=\"A box\u00a0[latex]E[\/latex] where the projection\u00a0[latex]D[\/latex] in the\u00a0[latex]xy[\/latex]-plane is of Type I.\" width=\"391\" height=\"391\" \/><figcaption id=\"caption-attachment-1389\" class=\"wp-caption-text\">Figure 2. Cylindrical coordinates are similar to polar coordinates with a vertical [latex]z[\/latex] coordinate added.<\/figcaption><\/figure>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p class=\"whitespace-normal break-words\"><strong>From rectangular to cylindrical:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]{x} = {r}{\\cos}{\\theta}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]{y} = {r}{\\sin}{\\theta}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z = z[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>From cylindrical to rectangular:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]{r^2} = {x^2} + {y^2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]{\\theta} = {\\tan^{-1}}{({\\frac{y}{x}})}[\/latex] (note: you may need to add [latex]\\pi[\/latex] to reach the correct quadrant)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z = z[\/latex]<\/li>\n<\/ul>\n<\/section>\n<p class=\"whitespace-normal break-words\">In the two-dimensional plane with a rectangular coordinate system, when we say [latex]x=k[\/latex] (constant) we mean an unbounded vertical line parallel to the [latex]y[\/latex]-axis and when [latex]y=l[\/latex] (constant) we mean an unbounded horizontal line parallel to the [latex]x[\/latex]-axis. With the polar coordinate system, when we say [latex]r=c[\/latex] (constant), we mean a circle of radius [latex]c[\/latex] units and when [latex]{\\theta} = {\\alpha}[\/latex] (constant) we mean an infinite ray making an angle [latex]{\\alpha}[\/latex] with the positive [latex]x[\/latex]-axis.<\/p>\n<p class=\"whitespace-normal break-words\">Similarly, in three-dimensional space with rectangular coordinates [latex](x, y, z)[\/latex], the equations [latex]x=k[\/latex], [latex]y=l[\/latex], and [latex]z=m[\/latex], where [latex]k[\/latex], [latex]l[\/latex], and [latex]m[\/latex] are constants, represent unbounded planes parallel to the [latex]yz[\/latex]-plane, [latex]xz[\/latex]-plane, and [latex]xy[\/latex]-plane, respectively.<\/p>\n<p class=\"whitespace-normal break-words\">With cylindrical coordinates [latex]{({r},{\\theta},{z})}[\/latex], the equations [latex]{r} = {c}[\/latex], [latex]{\\theta} = {\\alpha}[\/latex], and [latex]z=m[\/latex], where [latex]{c}[\/latex], [latex]{\\alpha}[\/latex], and [latex]m[\/latex] are constants, represent:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]r = c[\/latex]: an unbounded vertical cylinder with the [latex]z[\/latex]-axis as its radial axis<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\theta = \\alpha[\/latex]: a plane making a constant angle [latex]{\\alpha}[\/latex] with the [latex]xy[\/latex]-plane<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z = m[\/latex]: an unbounded horizontal plane parallel to the [latex]xy[\/latex]-plane<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">This means that the circular cylinder [latex]x^{2}+y^{2}=c^{2}[\/latex] in rectangular coordinates can be represented simply as [latex]r=c[\/latex] in cylindrical coordinates.<\/p>\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":11,"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-716","chapter","type-chapter","status-publish","hentry"],"part":434,"module-header":"learn_it","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\/716","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\/716\/revisions"}],"predecessor-version":[{"id":1935,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/716\/revisions\/1935"}],"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\/716\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=716"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=716"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=716"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=716"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}