{"id":1305,"date":"2025-10-17T13:23:54","date_gmt":"2025-10-17T13:23:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=1305"},"modified":"2025-11-04T14:30:04","modified_gmt":"2025-11-04T14:30:04","slug":"change-of-variables-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/change-of-variables-learn-it-4\/","title":{"raw":"Change of Variables: Learn It 4","rendered":"Change of Variables: Learn It 4"},"content":{"raw":"<h2 data-type=\"title\">Change of Variables for Triple Integrals<\/h2>\r\n<p class=\"whitespace-normal break-words\">Changing variables in triple integrals works exactly the same way as for double integrals. Cylindrical and spherical coordinate substitutions are special cases of this general method.<\/p>\r\n<p class=\"whitespace-normal break-words\">Suppose [latex]G[\/latex] is a region in [latex]uvw[\/latex]-space that maps to region [latex]D[\/latex] in [latex]xyz[\/latex]-space (Figure 6) by a one-to-one [latex]C^1[\/latex] transformation [latex]T(u,v,w) = (x,y,z)[\/latex] where:<\/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 = g(u,v,w)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y = h(u,v,w)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]z = k(u,v,w)[\/latex]<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_1427\" align=\"aligncenter\" width=\"930\"]<img class=\"size-full wp-image-1427\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/28043418\/5-7-10.jpeg\" alt=\"On the left-hand side of this figure, there is a region G in u v w space. Then there is an arrow from this graph to the right-hand side of the figure marked with x = g(u, v, w), y = h(u, v, w), and z = k(u, v, w). On the right-hand side of this figure there is a region D in xyz space.\" width=\"930\" height=\"350\" \/> Figure 6. A region [latex]G[\/latex] in [latex]uvw[\/latex]-space mapped to a region\u00a0[latex]D[\/latex] in [latex]xyz[\/latex]-space.[\/caption]\r\n<p class=\"whitespace-normal break-words\">Any function [latex]F(x,y,z)[\/latex] defined on [latex]D[\/latex] can be expressed as a function [latex]H(u,v,w)[\/latex] defined on [latex]G[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]F(x,y,z)=F(g(u,v,w), \\ h(u,v,w) \\ k(u,v,w))= \\ H(u,v,w)[\/latex].<\/p>\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p class=\"whitespace-normal break-words\"><strong>Two-Dimensional Jacobian<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">For a transformation [latex]T(u,v) = (x,y)[\/latex], the Jacobian is the [latex]2 \\times 2[\/latex] determinant:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]J(u,v) = \\begin{vmatrix} \\dfrac{\\partial x}{\\partial u} &amp; \\dfrac{\\partial y}{\\partial u} \\\\ \\dfrac{\\partial x}{\\partial v} &amp; \\dfrac{\\partial y}{\\partial v} \\end{vmatrix}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">The three-dimensional case extends this to a [latex]3 \\times 3[\/latex] determinant.<\/p>\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">The <strong>Jacobian determinant<\/strong> [latex]J(u,v,w)[\/latex] in three variables is defined as:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]J(u,v,w) = \\begin{vmatrix} \\dfrac{\\partial x}{\\partial u} &amp; \\dfrac{\\partial y}{\\partial u} &amp; \\dfrac{\\partial z}{\\partial u} \\\\ \\dfrac{\\partial x}{\\partial v} &amp; \\dfrac{\\partial y}{\\partial v} &amp; \\dfrac{\\partial z}{\\partial v} \\\\ \\dfrac{\\partial x}{\\partial w} &amp; \\dfrac{\\partial y}{\\partial w} &amp; \\dfrac{\\partial z}{\\partial w} \\end{vmatrix}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">This can also be written equivalently as:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]J(u,v,w) = \\begin{vmatrix} \\dfrac{\\partial x}{\\partial u} &amp; \\dfrac{\\partial x}{\\partial v} &amp; \\dfrac{\\partial x}{\\partial w} \\\\ \\dfrac{\\partial y}{\\partial u} &amp; \\dfrac{\\partial y}{\\partial v} &amp; \\dfrac{\\partial y}{\\partial w} \\\\ \\dfrac{\\partial z}{\\partial u} &amp; \\dfrac{\\partial z}{\\partial v} &amp; \\dfrac{\\partial z}{\\partial w} \\end{vmatrix}[\/latex]<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>jacobian determinant in three variables<\/h3>\r\n<p class=\"whitespace-normal break-words\">For transformation [latex]T(u,v,w) = (x,y,z)[\/latex]:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]J(u,v,w) = \\dfrac{\\partial(x,y,z)}{\\partial(u,v,w)} = \\begin{vmatrix} \\dfrac{\\partial x}{\\partial u} &amp; \\dfrac{\\partial x}{\\partial v} &amp; \\dfrac{\\partial x}{\\partial w} \\\\ \\dfrac{\\partial y}{\\partial u} &amp; \\dfrac{\\partial y}{\\partial v} &amp; \\dfrac{\\partial y}{\\partial w} \\\\ \\dfrac{\\partial z}{\\partial u} &amp; \\dfrac{\\partial z}{\\partial v} &amp; \\dfrac{\\partial z}{\\partial w} \\end{vmatrix}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">This determinant measures how the transformation scales volumes.<\/p>\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">The Jacobian is commonly denoted simply as [latex]\\dfrac{\\partial(x,y,z)}{\\partial(u,v,w)}[\/latex].<\/p>\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"whitespace-normal break-words\">Just as with two variables, you can arrange the [latex]3 \\times 3[\/latex] determinant in two equivalent ways:<\/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\"><strong>By output variables<\/strong>: Rows for [latex]x[\/latex], [latex]y[\/latex], [latex]z[\/latex] derivatives<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>By input variables<\/strong>: Columns for [latex]u[\/latex], [latex]v[\/latex], [latex]w[\/latex] derivatives<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-pre-wrap break-words\">Both give the same result when you compute the determinant.<\/p>\r\n\r\n<\/section>With the transformation and Jacobian defined, we can state the theorem for changing variables in triple integrals.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>change of variables for triple integrals<\/h3>\r\n<p class=\"whitespace-normal break-words\">Let [latex]T(u,v,w) = (x,y,z)[\/latex] where [latex]x = g(u,v,w)[\/latex], [latex]y = h(u,v,w)[\/latex], and [latex]z = k(u,v,w)[\/latex] be a one-to-one [latex]C^1[\/latex] transformation with a nonzero Jacobian that maps region [latex]G[\/latex] in [latex]uvw[\/latex]-space into region [latex]D[\/latex] in [latex]xyz[\/latex]-space. If [latex]F[\/latex] is continuous on [latex]D[\/latex], then:<\/p>\r\n[latex]\\hspace{3cm}\\begin{align}\r\n\r\n\\underset{R}{\\displaystyle\\iiint}F(x,y,z)dV&amp;=\\underset{G}{\\displaystyle\\iiint}F(g(u,v,w), \\ h(u,v,w), \\ k(u,v,w))\\Bigg|\\frac{\\partial(x,y,z)}{\\partial(u,v,w)}\\Bigg|du \\ dv \\ dw \\\\\r\n\r\n&amp;=\\underset{G}{\\displaystyle\\iiint}H(u,v,w)|J(u,v,w)|du \\ dv \\ dw\r\n\r\n\\end{align}[\/latex].\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">In other words, when changing variables from [latex](x,y,z)[\/latex] to [latex](u,v,w)[\/latex]:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]dV = dx \\, dy \\, dz = | \\dfrac{\\partial(x,y,z)}{\\partial(u,v,w)} | du \\, dv \\, dw[\/latex]<\/p>\r\n\r\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">\r\n<p class=\"whitespace-normal break-words\"><strong>Cylindrical and Spherical Coordinates<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">The coordinate systems you've already used are special cases of this general theorem:<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Cylindrical coordinates<\/strong>: [latex]x = r\\cos\\theta[\/latex], [latex]y = r\\sin\\theta[\/latex], [latex]z = z[\/latex]<\/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\">Jacobian: [latex]J(r,\\theta,z) = r[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Volume element: [latex]dV = r , dr , d\\theta , dz[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Spherical coordinates<\/strong>: [latex]x = \\rho\\sin\\phi\\cos\\theta[\/latex], [latex]y = \\rho\\sin\\phi\\sin\\theta[\/latex], [latex]z = \\rho\\cos\\phi[\/latex]<\/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\">Jacobian: [latex]J(\\rho,\\phi,\\theta) = \\rho^2\\sin\\phi[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Volume element: [latex]dV = \\rho^2\\sin\\phi , d\\rho , d\\phi , d\\theta[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-pre-wrap break-words\">We'll verify these formulas using the change of variables theorem.<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1167793293911\">Derive the formula in triple integrals for<\/p>\r\n\r\n<ol id=\"fs-id1167793293914\" type=\"a\">\r\n \t<li>cylindrical and<\/li>\r\n \t<li>spherical coordinates.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"827454432\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"827454432\"]\r\n<ol id=\"fs-id1167793498790\" type=\"a\">\r\n \t<li>For cylindrical coordinates, the transformation is [latex]T(r,\\theta,z)=(x,y,z)[\/latex] from the Cartesian [latex]r\\theta{z}[\/latex]--plane\u00a0\u00a0to the Cartesian [latex]xyz[\/latex]-plane\u00a0(Figure 7). Here [latex]x=r\\cos\\theta[\/latex], [latex]y=r\\sin\\theta[\/latex], and [latex]z=z[\/latex]. The Jacobian for the transformation is<span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\hspace{1.5cm}\\large{\\begin{align}J(r,\\theta,z)&amp;=\\frac{\\partial(x,y,z)}{\\partial(r,\\theta,z)}=\\begin{vmatrix}\\frac{\\partial{x}}{\\partial{r}}&amp;\\frac{\\partial{x}}{\\partial{\\theta}}&amp;\\frac{\\partial{x}}{\\partial{z}} \\\\ \\frac{\\partial{y}}{\\partial{r}}&amp;\\frac{\\partial{y}}{\\partial{\\theta}}&amp;\\frac{\\partial{y}}{\\partial{z}} \\\\ \\frac{\\partial{z}}{\\partial{r}}&amp;\\frac{\\partial{z}}{\\partial{\\theta}}&amp;\\frac{\\partial{z}}{\\partial{z}}\\end{vmatrix} \\\\&amp;=\\begin{vmatrix}\\cos\\theta&amp;-r\\sin\\theta&amp;0 \\\\ \\sin\\theta&amp;r\\cos\\theta&amp;0 \\\\ 0&amp;0&amp;1\\end{vmatrix}=r\\cos^2\\theta+r\\sin^2\\theta=r(\\cos^2\\theta+\\sin^s\\theta)=r \\end{align}}[\/latex].\r\n<span data-type=\"newline\">\r\n<\/span>We know that [latex]r\\geq0[\/latex], so [latex]|J(r,\\theta,z)|=r[\/latex]. Then the triple integral is<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1167793640526\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\large{\\underset{D}{\\displaystyle\\iiint}f(x,y,z)dV=\\underset{G}{\\displaystyle\\iiint}f(r\\cos\\theta,r\\sin\\theta,z)r \\ dr \\ d\\theta \\ dz}[\/latex].<\/div>\r\n[caption id=\"attachment_1428\" align=\"aligncenter\" width=\"898\"]<img class=\"size-full wp-image-1428\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/28043458\/5-7-11.jpeg\" alt=\"On the left-hand side of this figure, there is a cube G with sides parallel to the coordinate axes in cylindrical coordinate space. Then there is an arrow from this graph to the right-hand side of the figure marked with x = r cos theta, y = r sin theta, and z = z. On the right-hand side of this figure there is a region D in x y z space that is a thick annulus. The top is labeled z = constant, the flat vertical side is labeled theta = constant, and the outermost side is labeled r = constant.\" width=\"898\" height=\"391\" \/> Figure 7. The transformation from rectangular coordinates to cylindrical coordinates can be treated as a change of variables from region [latex]G[\/latex] in [latex]r\\theta{z}[\/latex]-space to region\u00a0[latex]D[\/latex] in [latex]xyz[\/latex]-space.[\/caption]<\/li>\r\n \t<li>For spherical coordinates, the transformation is [latex]T(\\rho,\\theta,\\varphi)=(x,y,z)[\/latex] from the Cartesian [latex]p\\theta\\varphi[\/latex]--plane\u00a0\u00a0to the Cartesian [latex]xyz[\/latex]--plane\u00a0\u00a0(Figure 8). Here [latex]x=\\rho\\sin\\varphi\\cos\\theta[\/latex], [latex]y=\\rho\\sin\\varphi\\sin\\theta[\/latex], and [latex]z=\\rho\\cos\\varphi[\/latex]. The Jacobian for the transformation is<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1167793373656\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\large{J(\\rho,\\theta,\\varphi)=\\frac{\\partial{(x,y,z)}}{\\partial{(\\rho,\\theta\\varphi)}}=\\begin{vmatrix}\\frac{\\partial{x}}{\\partial{\\rho}}&amp;\\frac{\\partial{x}}{\\partial{\\theta}}&amp;\\frac{\\partial{x}}{\\partial{\\varphi}} \\\\ \\frac{\\partial{y}}{\\partial{\\rho}}&amp;\\frac{\\partial{y}}{\\partial{\\theta}}&amp;\\frac{\\partial{y}}{\\partial{\\varphi}} \\\\ \\frac{\\partial{z}}{\\partial{\\rho}}&amp;\\frac{\\partial{z}}{\\partial{\\theta}}&amp;\\frac{\\partial{z}}{\\partial{\\varphi}}\\end{vmatrix}=\\begin{vmatrix}\\sin\\varphi\\cos\\theta&amp;-\\rho\\sin\\varphi\\sin\\theta&amp;\\rho\\cos\\varphi\\cos\\theta \\\\ \\sin\\varphi\\sin\\theta&amp;-\\rho\\sin\\varphi\\cos\\theta&amp;\\rho\\cos\\varphi\\sin\\theta \\\\ \\cos\\theta&amp;0&amp;-\\rho\\sin\\varphi\\end{vmatrix}}[\/latex]<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>Expanding the determinant with respect to the third row:<span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\hspace{2cm}\\begin{align}\r\n\r\n&amp;=\\cos\\varphi\\begin{vmatrix}-\\rho\\sin\\varphi\\sin\\theta&amp;\\rho\\cos\\varphi\\cos\\theta \\\\ \\rho\\sin\\varphi\\sin\\theta&amp;\\rho\\cos\\varphi\\sin\\theta\\end{vmatrix}-\\rho\\sin\\varphi\\begin{vmatrix}\\sin\\varphi\\cos\\theta&amp;-\\rho\\sin\\varphi\\sin\\theta \\\\ \\sin\\varphi\\sin\\theta&amp;\\rho\\sin\\varphi\\cos\\theta\\end{vmatrix} \\\\\r\n\r\n&amp;=\\cos\\varphi(-\\rho^2\\sin\\varphi\\cos\\varphi\\sin^2\\theta-\\rho^2\\sin\\varphi\\cos\\varphi\\cos^2\\theta)-\\rho\\sin\\varphi(\\rho\\sin^2\\varphi\\cos^2\\theta+\\rho\\sin^2\\varphi\\sin^2\\theta) \\\\\r\n\r\n&amp;=-\\rho^2\\sin\\varphi\\cos^2\\varphi(\\sin^2\\theta+\\cos^2\\theta)-\\rho^2\\sin\\varphi\\sin^2\\varphi(\\sin^2\\theta+\\cos^2\\theta) \\\\\r\n\r\n&amp;=-\\rho^2\\sin\\varphi\\cos^2\\varphi-\\rho^2\\sin\\varphi\\sin^2\\varphi \\\\\r\n\r\n&amp;=-\\rho^2\\sin\\varphi(\\cos^2\\varphi+\\sin^2\\varphi)=-\\rho^2\\sin\\varphi\\end{align}[\/latex]\r\n<span data-type=\"newline\">\r\n<\/span>Since [latex]0\\leq\\varphi\\leq\\pi[\/latex], we must have [latex]\\sin\\varphi\\geq0[\/latex]. Thus [latex]|J(\\rho,\\theta,\\varphi)|=|-\\rho^2\\sin\\varphi|=\\rho^2\\sin\\varphi[\/latex].\r\n<div id=\"CNX_Calc_Figure_15_07_012\" class=\"os-figure\">\r\n<div class=\"os-caption-container\">[caption id=\"attachment_1429\" align=\"aligncenter\" width=\"895\"]<img class=\"size-full wp-image-1429\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/28043542\/5-7-12.jpeg\" alt=\"On the left-hand side of this figure, there is a cube G with sides parallel to the coordinate axes in rho phi theta space. Then there is an arrow from this graph to the right-hand side of the figure marked with x = rho sin phi cos theta, y = rho sin phi sin theta, and z = rho cos phi. On the right-hand side of this figure there is a region D in xyz space that is a thick annulus and has the point (x, y, z) shown as being equal to (rho, phi, theta). The top is labeled phi = constant, the flat vertical side is labeled theta = constant, and the outermost side is labeled rho = constant.\" width=\"895\" height=\"391\" \/> Figure 8. The transformation from rectangular coordinates to cylindrical coordinates can be treated as a change of variables from region [latex]G[\/latex] in [latex]\\rho\\theta\\varphi[\/latex]-space to region\u00a0[latex]D[\/latex] in [latex]xyz[\/latex]-space.[\/caption]<\/div>\r\n<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>Then the triple integral becomes<span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\underset{D}{\\displaystyle\\iiint}f(x,y,z)dV=\\underset{G}{\\displaystyle\\iiint}f(\\rho\\sin\\varphi\\cos\\theta,\\rho\\sin\\varphi\\sin\\theta,\\rho\\cos\\varphi)\\rho^2\\sin\\varphi \\ d\\rho \\ d\\varphi \\ d\\theta[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1167793376275\">Evaluate the triple integral<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\int_0^3\\displaystyle\\int_0^4\\displaystyle\\int_{y\/2}^{(y\/2)+1}\\left(x+\\frac{z}3\\right)dx \\ dy \\ dz}[\/latex]<\/p>\r\n<p id=\"fs-id1167793638297\">in [latex]xyz[\/latex]-space by using the transformation<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{u=(2x-y)\/2, \\ v=y\/2,\\text{ and }w=z\/3}[\/latex].<\/p>\r\n<p id=\"fs-id1167793930128\">Then integrate over an appropriate region in [latex]uvw[\/latex]-space<\/p>\r\n[reveal-answer q=\"458273487\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"458273487\"]\r\n<p id=\"fs-id1167793930147\">As before, some kind of sketch of the region [latex]G[\/latex] in [latex]xyz[\/latex]-space over which we have to perform the integration can help identify the region [latex]D[\/latex] in [latex]uvw[\/latex]-space (Figure 9). Clearly [latex]G[\/latex] in [latex]xyz[\/latex]-space is bounded by the planes [latex]x=y\/2[\/latex], [latex]x=(y\/2)+1[\/latex], [latex]y=0[\/latex],[latex]y=4[\/latex], [latex]z=0[\/latex], and [latex]z=4[\/latex]. We also know that we have to use [latex]u=(2x-y)\/2[\/latex], [latex]v=y\/2[\/latex] and [latex]w=z\/3[\/latex] for the transformations. We need to solve for [latex]x[\/latex], [latex]y[\/latex], and [latex]z[\/latex]. Here we find that\u00a0[latex]x=u+v[\/latex],\u00a0[latex]y=2v[\/latex], and\u00a0[latex]z=3w[\/latex].<\/p>\r\n<p id=\"fs-id1167794223120\">Using elementary algebra, we can find the corresponding surfaces for the region [latex]G[\/latex] and the limits of integration in [latex]uvw[\/latex]-space. It is convenient to list these equations in a table.<\/p>\r\n\r\n<div id=\"fs-id1167794223141\" class=\"os-table\">\r\n<table class=\"unnumbered\" style=\"height: 166px; width: 622px;\" data-id=\"fs-id1167794223141\" data-label=\"\">\r\n<thead>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<th style=\"height: 24px; width: 259.734px;\" scope=\"col\" data-valign=\"top\" data-align=\"left\">Equations in [latex]xyz[\/latex] for the region [latex]D[\/latex]<\/th>\r\n<th style=\"height: 24px; width: 204.016px;\" scope=\"col\" data-valign=\"top\" data-align=\"left\">Corresponding equations in [latex]uvw[\/latex] for the region [latex]G[\/latex]<\/th>\r\n<th style=\"height: 24px; width: 123.219px;\" scope=\"col\" data-valign=\"top\" data-align=\"left\">Limits for the integration in [latex]uvw[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"height: 22px; width: 259.734px;\" data-valign=\"top\" data-align=\"left\">[latex]x=y\/2[\/latex]<\/td>\r\n<td style=\"height: 22px; width: 204.016px;\" data-valign=\"top\" data-align=\"left\">[latex]u+v=2v\/2=v[\/latex]<\/td>\r\n<td style=\"height: 22px; width: 123.219px;\" data-valign=\"top\" data-align=\"left\">[latex]u=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"height: 24px; width: 259.734px;\" data-valign=\"top\" data-align=\"left\">[latex]x=y\/2[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 204.016px;\" data-valign=\"top\" data-align=\"left\">[latex]u+v=(2v\/2)+1=v+1[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 123.219px;\" data-valign=\"top\" data-align=\"left\">[latex]u=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"height: 24px; width: 259.734px;\" data-valign=\"top\" data-align=\"left\">[latex]y=0[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 204.016px;\" data-valign=\"top\" data-align=\"left\">[latex]2v=0[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 123.219px;\" data-valign=\"top\" data-align=\"left\">[latex]v=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"height: 24px; width: 259.734px;\" data-valign=\"top\" data-align=\"left\">[latex]y=4[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 204.016px;\" data-valign=\"top\" data-align=\"left\">[latex]2v=4[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 123.219px;\" data-valign=\"top\" data-align=\"left\">[latex]v=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"height: 24px; width: 259.734px;\" data-valign=\"top\" data-align=\"left\">[latex]z=0[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 204.016px;\" data-valign=\"top\" data-align=\"left\">[latex]3w=0[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 123.219px;\" data-valign=\"top\" data-align=\"left\">[latex]w=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\" valign=\"top\">\r\n<td style=\"height: 24px; width: 259.734px;\" data-valign=\"top\" data-align=\"left\">[latex]z=3[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 204.016px;\" data-valign=\"top\" data-align=\"left\">[latex]3w=3[\/latex]<\/td>\r\n<td style=\"height: 24px; width: 123.219px;\" data-valign=\"top\" data-align=\"left\">[latex]w=1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"CNX_Calc_Figure_15_07_013\" class=\"os-figure\">\r\n<div class=\"os-caption-container\">[caption id=\"attachment_1983\" align=\"aligncenter\" width=\"894\"]<img class=\"size-full wp-image-1983\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/04142142\/9ac91afe22764bb4840cdcc0310bae65f7605c69.webp\" alt=\"On the left-hand side of this figure, there is a box G with sides 1, 2, and 1 along the u, v, and w axes, respectively. Then there is an arrow from this graph to the right-hand side of the figure marked with x = u + v, y = 2v, and z = 3w. On the right-hand side of this figure there is a region D in xyz space that is a rotated box with sides 1, 4, and 3 along the x, y, and z axes. The rear plane is marked x = y\/2 or y = 2x. The front plane is marked x = y\/2 + 1 or y = 2x minus 2.\" width=\"894\" height=\"435\" \/> Figure 9. The region [latex]G[\/latex] in [latex]uvw[\/latex]-space is transformed to region D in [latex]xyz[\/latex]-space.[\/caption]<\/div>\r\n<div><\/div>\r\n<\/div>\r\n<p id=\"fs-id1167793607338\">Now we can calculate the Jacobian for the transformation:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{J(u,v,w)=\\begin{vmatrix}\\frac{\\partial{x}}{\\partial{u}}&amp;\\frac{\\partial{x}}{\\partial{v}}&amp;\\frac{\\partial{x}}{\\partial{w}} \\\\ \\frac{\\partial{y}}{\\partial{u}}&amp;\\frac{\\partial{y}}{\\partial{v}}&amp;\\frac{\\partial{y}}{\\partial{w}} \\\\ \\frac{\\partial{z}}{\\partial{u}}&amp;\\frac{\\partial{z}}{\\partial{v}}&amp;\\frac{\\partial{z}}{\\partial{w}}\\end{vmatrix}=\\begin{vmatrix}1&amp;1&amp;0 \\\\ 0&amp;2&amp;0 \\\\ 0&amp;0&amp;3\\end{vmatrix}=6}[\/latex].<\/p>\r\n<p id=\"fs-id1167794296460\">The function to be integrated becomes<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{f(x,y,z)=x+\\frac{z}3=u+v+\\frac{3w}3=u+v+w}[\/latex].<\/p>\r\n<p id=\"fs-id1167793420356\">We are now ready to put everything together and complete the problem.<\/p>\r\n[latex]\\hspace{3cm}\\large{\\begin{align}\r\n\r\n&amp;\\hspace{1cm}\\displaystyle\\int_0^3\\displaystyle\\int_0^4\\displaystyle\\int_{y\/2}^{(y\/2)+1}\\left(x+\\frac{z}3\\right)dx \\ dy \\ dz \\\\\r\n\r\n&amp;=\\displaystyle\\int_0^1\\displaystyle\\int_0^2\\displaystyle\\int_0^1(u+v+w)|J(u,v,w)|du \\ dv \\ dw=\\displaystyle\\int_0^1\\displaystyle\\int_0^2\\displaystyle\\int_0^1(u+v+w)|6|du \\ dv \\ dw \\\\\r\n\r\n&amp;=6\\displaystyle\\int_0^1\\displaystyle\\int_0^2\\displaystyle\\int_0^1(u+v+w)du \\ dv \\ dw=6\\displaystyle\\int_0^1\\displaystyle\\int_0^2\\left[\\frac{u^2}2+vu+wu\\right]_0^1dv \\ dw \\\\\r\n\r\n&amp;=6\\displaystyle\\int_0^1\\displaystyle\\int_0^2\\left(\\frac12+v+w\\right)dv \\ dw=6\\displaystyle\\int_0^1\\left[\\frac12v+\\frac{v^2}2+wv\\right]_0^2dw \\\\\r\n\r\n&amp;=6\\displaystyle\\int_0^1(3+2w)dw=6\\left[3w+w^2\\right]_0^1=24\r\n\r\n\\end{align}}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<h2 data-type=\"title\">Change of Variables for Triple Integrals<\/h2>\n<p class=\"whitespace-normal break-words\">Changing variables in triple integrals works exactly the same way as for double integrals. Cylindrical and spherical coordinate substitutions are special cases of this general method.<\/p>\n<p class=\"whitespace-normal break-words\">Suppose [latex]G[\/latex] is a region in [latex]uvw[\/latex]-space that maps to region [latex]D[\/latex] in [latex]xyz[\/latex]-space (Figure 6) by a one-to-one [latex]C^1[\/latex] transformation [latex]T(u,v,w) = (x,y,z)[\/latex] where:<\/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 = g(u,v,w)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y = h(u,v,w)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]z = k(u,v,w)[\/latex]<\/li>\n<\/ul>\n<figure id=\"attachment_1427\" aria-describedby=\"caption-attachment-1427\" style=\"width: 930px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1427\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/28043418\/5-7-10.jpeg\" alt=\"On the left-hand side of this figure, there is a region G in u v w space. Then there is an arrow from this graph to the right-hand side of the figure marked with x = g(u, v, w), y = h(u, v, w), and z = k(u, v, w). On the right-hand side of this figure there is a region D in xyz space.\" width=\"930\" height=\"350\" \/><figcaption id=\"caption-attachment-1427\" class=\"wp-caption-text\">Figure 6. A region [latex]G[\/latex] in [latex]uvw[\/latex]-space mapped to a region\u00a0[latex]D[\/latex] in [latex]xyz[\/latex]-space.<\/figcaption><\/figure>\n<p class=\"whitespace-normal break-words\">Any function [latex]F(x,y,z)[\/latex] defined on [latex]D[\/latex] can be expressed as a function [latex]H(u,v,w)[\/latex] defined on [latex]G[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]F(x,y,z)=F(g(u,v,w), \\ h(u,v,w) \\ k(u,v,w))= \\ H(u,v,w)[\/latex].<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p class=\"whitespace-normal break-words\"><strong>Two-Dimensional Jacobian<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">For a transformation [latex]T(u,v) = (x,y)[\/latex], the Jacobian is the [latex]2 \\times 2[\/latex] determinant:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]J(u,v) = \\begin{vmatrix} \\dfrac{\\partial x}{\\partial u} & \\dfrac{\\partial y}{\\partial u} \\\\ \\dfrac{\\partial x}{\\partial v} & \\dfrac{\\partial y}{\\partial v} \\end{vmatrix}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">The three-dimensional case extends this to a [latex]3 \\times 3[\/latex] determinant.<\/p>\n<\/section>\n<p class=\"whitespace-normal break-words\">The <strong>Jacobian determinant<\/strong> [latex]J(u,v,w)[\/latex] in three variables is defined as:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]J(u,v,w) = \\begin{vmatrix} \\dfrac{\\partial x}{\\partial u} & \\dfrac{\\partial y}{\\partial u} & \\dfrac{\\partial z}{\\partial u} \\\\ \\dfrac{\\partial x}{\\partial v} & \\dfrac{\\partial y}{\\partial v} & \\dfrac{\\partial z}{\\partial v} \\\\ \\dfrac{\\partial x}{\\partial w} & \\dfrac{\\partial y}{\\partial w} & \\dfrac{\\partial z}{\\partial w} \\end{vmatrix}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">This can also be written equivalently as:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]J(u,v,w) = \\begin{vmatrix} \\dfrac{\\partial x}{\\partial u} & \\dfrac{\\partial x}{\\partial v} & \\dfrac{\\partial x}{\\partial w} \\\\ \\dfrac{\\partial y}{\\partial u} & \\dfrac{\\partial y}{\\partial v} & \\dfrac{\\partial y}{\\partial w} \\\\ \\dfrac{\\partial z}{\\partial u} & \\dfrac{\\partial z}{\\partial v} & \\dfrac{\\partial z}{\\partial w} \\end{vmatrix}[\/latex]<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>jacobian determinant in three variables<\/h3>\n<p class=\"whitespace-normal break-words\">For transformation [latex]T(u,v,w) = (x,y,z)[\/latex]:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]J(u,v,w) = \\dfrac{\\partial(x,y,z)}{\\partial(u,v,w)} = \\begin{vmatrix} \\dfrac{\\partial x}{\\partial u} & \\dfrac{\\partial x}{\\partial v} & \\dfrac{\\partial x}{\\partial w} \\\\ \\dfrac{\\partial y}{\\partial u} & \\dfrac{\\partial y}{\\partial v} & \\dfrac{\\partial y}{\\partial w} \\\\ \\dfrac{\\partial z}{\\partial u} & \\dfrac{\\partial z}{\\partial v} & \\dfrac{\\partial z}{\\partial w} \\end{vmatrix}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">This determinant measures how the transformation scales volumes.<\/p>\n<\/section>\n<p class=\"whitespace-normal break-words\">The Jacobian is commonly denoted simply as [latex]\\dfrac{\\partial(x,y,z)}{\\partial(u,v,w)}[\/latex].<\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"whitespace-normal break-words\">Just as with two variables, you can arrange the [latex]3 \\times 3[\/latex] determinant in two equivalent ways:<\/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\"><strong>By output variables<\/strong>: Rows for [latex]x[\/latex], [latex]y[\/latex], [latex]z[\/latex] derivatives<\/li>\n<li class=\"whitespace-normal break-words\"><strong>By input variables<\/strong>: Columns for [latex]u[\/latex], [latex]v[\/latex], [latex]w[\/latex] derivatives<\/li>\n<\/ul>\n<p class=\"whitespace-pre-wrap break-words\">Both give the same result when you compute the determinant.<\/p>\n<\/section>\n<p>With the transformation and Jacobian defined, we can state the theorem for changing variables in triple integrals.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>change of variables for triple integrals<\/h3>\n<p class=\"whitespace-normal break-words\">Let [latex]T(u,v,w) = (x,y,z)[\/latex] where [latex]x = g(u,v,w)[\/latex], [latex]y = h(u,v,w)[\/latex], and [latex]z = k(u,v,w)[\/latex] be a one-to-one [latex]C^1[\/latex] transformation with a nonzero Jacobian that maps region [latex]G[\/latex] in [latex]uvw[\/latex]-space into region [latex]D[\/latex] in [latex]xyz[\/latex]-space. If [latex]F[\/latex] is continuous on [latex]D[\/latex], then:<\/p>\n<p>[latex]\\hspace{3cm}\\begin{align}    \\underset{R}{\\displaystyle\\iiint}F(x,y,z)dV&=\\underset{G}{\\displaystyle\\iiint}F(g(u,v,w), \\ h(u,v,w), \\ k(u,v,w))\\Bigg|\\frac{\\partial(x,y,z)}{\\partial(u,v,w)}\\Bigg|du \\ dv \\ dw \\\\    &=\\underset{G}{\\displaystyle\\iiint}H(u,v,w)|J(u,v,w)|du \\ dv \\ dw    \\end{align}[\/latex].<\/p>\n<\/section>\n<p class=\"whitespace-normal break-words\">In other words, when changing variables from [latex](x,y,z)[\/latex] to [latex](u,v,w)[\/latex]:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]dV = dx \\, dy \\, dz = | \\dfrac{\\partial(x,y,z)}{\\partial(u,v,w)} | du \\, dv \\, dw[\/latex]<\/p>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">\n<p class=\"whitespace-normal break-words\"><strong>Cylindrical and Spherical Coordinates<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">The coordinate systems you&#8217;ve already used are special cases of this general theorem:<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Cylindrical coordinates<\/strong>: [latex]x = r\\cos\\theta[\/latex], [latex]y = r\\sin\\theta[\/latex], [latex]z = z[\/latex]<\/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\">Jacobian: [latex]J(r,\\theta,z) = r[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Volume element: [latex]dV = r , dr , d\\theta , dz[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Spherical coordinates<\/strong>: [latex]x = \\rho\\sin\\phi\\cos\\theta[\/latex], [latex]y = \\rho\\sin\\phi\\sin\\theta[\/latex], [latex]z = \\rho\\cos\\phi[\/latex]<\/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\">Jacobian: [latex]J(\\rho,\\phi,\\theta) = \\rho^2\\sin\\phi[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Volume element: [latex]dV = \\rho^2\\sin\\phi , d\\rho , d\\phi , d\\theta[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-pre-wrap break-words\">We&#8217;ll verify these formulas using the change of variables theorem.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1167793293911\">Derive the formula in triple integrals for<\/p>\n<ol id=\"fs-id1167793293914\" type=\"a\">\n<li>cylindrical and<\/li>\n<li>spherical coordinates.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q827454432\">Show Solution<\/button><\/p>\n<div id=\"q827454432\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1167793498790\" type=\"a\">\n<li>For cylindrical coordinates, the transformation is [latex]T(r,\\theta,z)=(x,y,z)[\/latex] from the Cartesian [latex]r\\theta{z}[\/latex]&#8211;plane\u00a0\u00a0to the Cartesian [latex]xyz[\/latex]-plane\u00a0(Figure 7). Here [latex]x=r\\cos\\theta[\/latex], [latex]y=r\\sin\\theta[\/latex], and [latex]z=z[\/latex]. The Jacobian for the transformation is<span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\hspace{1.5cm}\\large{\\begin{align}J(r,\\theta,z)&=\\frac{\\partial(x,y,z)}{\\partial(r,\\theta,z)}=\\begin{vmatrix}\\frac{\\partial{x}}{\\partial{r}}&\\frac{\\partial{x}}{\\partial{\\theta}}&\\frac{\\partial{x}}{\\partial{z}} \\\\ \\frac{\\partial{y}}{\\partial{r}}&\\frac{\\partial{y}}{\\partial{\\theta}}&\\frac{\\partial{y}}{\\partial{z}} \\\\ \\frac{\\partial{z}}{\\partial{r}}&\\frac{\\partial{z}}{\\partial{\\theta}}&\\frac{\\partial{z}}{\\partial{z}}\\end{vmatrix} \\\\&=\\begin{vmatrix}\\cos\\theta&-r\\sin\\theta&0 \\\\ \\sin\\theta&r\\cos\\theta&0 \\\\ 0&0&1\\end{vmatrix}=r\\cos^2\\theta+r\\sin^2\\theta=r(\\cos^2\\theta+\\sin^s\\theta)=r \\end{align}}[\/latex].<br \/>\n<span data-type=\"newline\"><br \/>\n<\/span>We know that [latex]r\\geq0[\/latex], so [latex]|J(r,\\theta,z)|=r[\/latex]. Then the triple integral is<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1167793640526\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\large{\\underset{D}{\\displaystyle\\iiint}f(x,y,z)dV=\\underset{G}{\\displaystyle\\iiint}f(r\\cos\\theta,r\\sin\\theta,z)r \\ dr \\ d\\theta \\ dz}[\/latex].<\/div>\n<figure id=\"attachment_1428\" aria-describedby=\"caption-attachment-1428\" style=\"width: 898px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1428\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/28043458\/5-7-11.jpeg\" alt=\"On the left-hand side of this figure, there is a cube G with sides parallel to the coordinate axes in cylindrical coordinate space. Then there is an arrow from this graph to the right-hand side of the figure marked with x = r cos theta, y = r sin theta, and z = z. On the right-hand side of this figure there is a region D in x y z space that is a thick annulus. The top is labeled z = constant, the flat vertical side is labeled theta = constant, and the outermost side is labeled r = constant.\" width=\"898\" height=\"391\" \/><figcaption id=\"caption-attachment-1428\" class=\"wp-caption-text\">Figure 7. The transformation from rectangular coordinates to cylindrical coordinates can be treated as a change of variables from region [latex]G[\/latex] in [latex]r\\theta{z}[\/latex]-space to region\u00a0[latex]D[\/latex] in [latex]xyz[\/latex]-space.<\/figcaption><\/figure>\n<\/li>\n<li>For spherical coordinates, the transformation is [latex]T(\\rho,\\theta,\\varphi)=(x,y,z)[\/latex] from the Cartesian [latex]p\\theta\\varphi[\/latex]&#8211;plane\u00a0\u00a0to the Cartesian [latex]xyz[\/latex]&#8211;plane\u00a0\u00a0(Figure 8). Here [latex]x=\\rho\\sin\\varphi\\cos\\theta[\/latex], [latex]y=\\rho\\sin\\varphi\\sin\\theta[\/latex], and [latex]z=\\rho\\cos\\varphi[\/latex]. The Jacobian for the transformation is<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1167793373656\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\large{J(\\rho,\\theta,\\varphi)=\\frac{\\partial{(x,y,z)}}{\\partial{(\\rho,\\theta\\varphi)}}=\\begin{vmatrix}\\frac{\\partial{x}}{\\partial{\\rho}}&\\frac{\\partial{x}}{\\partial{\\theta}}&\\frac{\\partial{x}}{\\partial{\\varphi}} \\\\ \\frac{\\partial{y}}{\\partial{\\rho}}&\\frac{\\partial{y}}{\\partial{\\theta}}&\\frac{\\partial{y}}{\\partial{\\varphi}} \\\\ \\frac{\\partial{z}}{\\partial{\\rho}}&\\frac{\\partial{z}}{\\partial{\\theta}}&\\frac{\\partial{z}}{\\partial{\\varphi}}\\end{vmatrix}=\\begin{vmatrix}\\sin\\varphi\\cos\\theta&-\\rho\\sin\\varphi\\sin\\theta&\\rho\\cos\\varphi\\cos\\theta \\\\ \\sin\\varphi\\sin\\theta&-\\rho\\sin\\varphi\\cos\\theta&\\rho\\cos\\varphi\\sin\\theta \\\\ \\cos\\theta&0&-\\rho\\sin\\varphi\\end{vmatrix}}[\/latex]<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span>Expanding the determinant with respect to the third row:<span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\hspace{2cm}\\begin{align}    &=\\cos\\varphi\\begin{vmatrix}-\\rho\\sin\\varphi\\sin\\theta&\\rho\\cos\\varphi\\cos\\theta \\\\ \\rho\\sin\\varphi\\sin\\theta&\\rho\\cos\\varphi\\sin\\theta\\end{vmatrix}-\\rho\\sin\\varphi\\begin{vmatrix}\\sin\\varphi\\cos\\theta&-\\rho\\sin\\varphi\\sin\\theta \\\\ \\sin\\varphi\\sin\\theta&\\rho\\sin\\varphi\\cos\\theta\\end{vmatrix} \\\\    &=\\cos\\varphi(-\\rho^2\\sin\\varphi\\cos\\varphi\\sin^2\\theta-\\rho^2\\sin\\varphi\\cos\\varphi\\cos^2\\theta)-\\rho\\sin\\varphi(\\rho\\sin^2\\varphi\\cos^2\\theta+\\rho\\sin^2\\varphi\\sin^2\\theta) \\\\    &=-\\rho^2\\sin\\varphi\\cos^2\\varphi(\\sin^2\\theta+\\cos^2\\theta)-\\rho^2\\sin\\varphi\\sin^2\\varphi(\\sin^2\\theta+\\cos^2\\theta) \\\\    &=-\\rho^2\\sin\\varphi\\cos^2\\varphi-\\rho^2\\sin\\varphi\\sin^2\\varphi \\\\    &=-\\rho^2\\sin\\varphi(\\cos^2\\varphi+\\sin^2\\varphi)=-\\rho^2\\sin\\varphi\\end{align}[\/latex]<br \/>\n<span data-type=\"newline\"><br \/>\n<\/span>Since [latex]0\\leq\\varphi\\leq\\pi[\/latex], we must have [latex]\\sin\\varphi\\geq0[\/latex]. Thus [latex]|J(\\rho,\\theta,\\varphi)|=|-\\rho^2\\sin\\varphi|=\\rho^2\\sin\\varphi[\/latex].<\/p>\n<div id=\"CNX_Calc_Figure_15_07_012\" class=\"os-figure\">\n<div class=\"os-caption-container\">\n<figure id=\"attachment_1429\" aria-describedby=\"caption-attachment-1429\" style=\"width: 895px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1429\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/28043542\/5-7-12.jpeg\" alt=\"On the left-hand side of this figure, there is a cube G with sides parallel to the coordinate axes in rho phi theta space. Then there is an arrow from this graph to the right-hand side of the figure marked with x = rho sin phi cos theta, y = rho sin phi sin theta, and z = rho cos phi. On the right-hand side of this figure there is a region D in xyz space that is a thick annulus and has the point (x, y, z) shown as being equal to (rho, phi, theta). The top is labeled phi = constant, the flat vertical side is labeled theta = constant, and the outermost side is labeled rho = constant.\" width=\"895\" height=\"391\" \/><figcaption id=\"caption-attachment-1429\" class=\"wp-caption-text\">Figure 8. The transformation from rectangular coordinates to cylindrical coordinates can be treated as a change of variables from region [latex]G[\/latex] in [latex]\\rho\\theta\\varphi[\/latex]-space to region\u00a0[latex]D[\/latex] in [latex]xyz[\/latex]-space.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span>Then the triple integral becomes<span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\underset{D}{\\displaystyle\\iiint}f(x,y,z)dV=\\underset{G}{\\displaystyle\\iiint}f(\\rho\\sin\\varphi\\cos\\theta,\\rho\\sin\\varphi\\sin\\theta,\\rho\\cos\\varphi)\\rho^2\\sin\\varphi \\ d\\rho \\ d\\varphi \\ d\\theta[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1167793376275\">Evaluate the triple integral<\/p>\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\int_0^3\\displaystyle\\int_0^4\\displaystyle\\int_{y\/2}^{(y\/2)+1}\\left(x+\\frac{z}3\\right)dx \\ dy \\ dz}[\/latex]<\/p>\n<p id=\"fs-id1167793638297\">in [latex]xyz[\/latex]-space by using the transformation<\/p>\n<p style=\"text-align: center;\">[latex]\\large{u=(2x-y)\/2, \\ v=y\/2,\\text{ and }w=z\/3}[\/latex].<\/p>\n<p id=\"fs-id1167793930128\">Then integrate over an appropriate region in [latex]uvw[\/latex]-space<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q458273487\">Show Solution<\/button><\/p>\n<div id=\"q458273487\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793930147\">As before, some kind of sketch of the region [latex]G[\/latex] in [latex]xyz[\/latex]-space over which we have to perform the integration can help identify the region [latex]D[\/latex] in [latex]uvw[\/latex]-space (Figure 9). Clearly [latex]G[\/latex] in [latex]xyz[\/latex]-space is bounded by the planes [latex]x=y\/2[\/latex], [latex]x=(y\/2)+1[\/latex], [latex]y=0[\/latex],[latex]y=4[\/latex], [latex]z=0[\/latex], and [latex]z=4[\/latex]. We also know that we have to use [latex]u=(2x-y)\/2[\/latex], [latex]v=y\/2[\/latex] and [latex]w=z\/3[\/latex] for the transformations. We need to solve for [latex]x[\/latex], [latex]y[\/latex], and [latex]z[\/latex]. Here we find that\u00a0[latex]x=u+v[\/latex],\u00a0[latex]y=2v[\/latex], and\u00a0[latex]z=3w[\/latex].<\/p>\n<p id=\"fs-id1167794223120\">Using elementary algebra, we can find the corresponding surfaces for the region [latex]G[\/latex] and the limits of integration in [latex]uvw[\/latex]-space. It is convenient to list these equations in a table.<\/p>\n<div id=\"fs-id1167794223141\" class=\"os-table\">\n<table class=\"unnumbered\" style=\"height: 166px; width: 622px;\" data-id=\"fs-id1167794223141\" data-label=\"\">\n<thead>\n<tr style=\"height: 24px;\" valign=\"top\">\n<th style=\"height: 24px; width: 259.734px;\" scope=\"col\" data-valign=\"top\" data-align=\"left\">Equations in [latex]xyz[\/latex] for the region [latex]D[\/latex]<\/th>\n<th style=\"height: 24px; width: 204.016px;\" scope=\"col\" data-valign=\"top\" data-align=\"left\">Corresponding equations in [latex]uvw[\/latex] for the region [latex]G[\/latex]<\/th>\n<th style=\"height: 24px; width: 123.219px;\" scope=\"col\" data-valign=\"top\" data-align=\"left\">Limits for the integration in [latex]uvw[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"height: 22px; width: 259.734px;\" data-valign=\"top\" data-align=\"left\">[latex]x=y\/2[\/latex]<\/td>\n<td style=\"height: 22px; width: 204.016px;\" data-valign=\"top\" data-align=\"left\">[latex]u+v=2v\/2=v[\/latex]<\/td>\n<td style=\"height: 22px; width: 123.219px;\" data-valign=\"top\" data-align=\"left\">[latex]u=0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"height: 24px; width: 259.734px;\" data-valign=\"top\" data-align=\"left\">[latex]x=y\/2[\/latex]<\/td>\n<td style=\"height: 24px; width: 204.016px;\" data-valign=\"top\" data-align=\"left\">[latex]u+v=(2v\/2)+1=v+1[\/latex]<\/td>\n<td style=\"height: 24px; width: 123.219px;\" data-valign=\"top\" data-align=\"left\">[latex]u=1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"height: 24px; width: 259.734px;\" data-valign=\"top\" data-align=\"left\">[latex]y=0[\/latex]<\/td>\n<td style=\"height: 24px; width: 204.016px;\" data-valign=\"top\" data-align=\"left\">[latex]2v=0[\/latex]<\/td>\n<td style=\"height: 24px; width: 123.219px;\" data-valign=\"top\" data-align=\"left\">[latex]v=0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"height: 24px; width: 259.734px;\" data-valign=\"top\" data-align=\"left\">[latex]y=4[\/latex]<\/td>\n<td style=\"height: 24px; width: 204.016px;\" data-valign=\"top\" data-align=\"left\">[latex]2v=4[\/latex]<\/td>\n<td style=\"height: 24px; width: 123.219px;\" data-valign=\"top\" data-align=\"left\">[latex]v=2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"height: 24px; width: 259.734px;\" data-valign=\"top\" data-align=\"left\">[latex]z=0[\/latex]<\/td>\n<td style=\"height: 24px; width: 204.016px;\" data-valign=\"top\" data-align=\"left\">[latex]3w=0[\/latex]<\/td>\n<td style=\"height: 24px; width: 123.219px;\" data-valign=\"top\" data-align=\"left\">[latex]w=0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\" valign=\"top\">\n<td style=\"height: 24px; width: 259.734px;\" data-valign=\"top\" data-align=\"left\">[latex]z=3[\/latex]<\/td>\n<td style=\"height: 24px; width: 204.016px;\" data-valign=\"top\" data-align=\"left\">[latex]3w=3[\/latex]<\/td>\n<td style=\"height: 24px; width: 123.219px;\" data-valign=\"top\" data-align=\"left\">[latex]w=1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"CNX_Calc_Figure_15_07_013\" class=\"os-figure\">\n<div class=\"os-caption-container\">\n<figure id=\"attachment_1983\" aria-describedby=\"caption-attachment-1983\" style=\"width: 894px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1983\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/04142142\/9ac91afe22764bb4840cdcc0310bae65f7605c69.webp\" alt=\"On the left-hand side of this figure, there is a box G with sides 1, 2, and 1 along the u, v, and w axes, respectively. Then there is an arrow from this graph to the right-hand side of the figure marked with x = u + v, y = 2v, and z = 3w. On the right-hand side of this figure there is a region D in xyz space that is a rotated box with sides 1, 4, and 3 along the x, y, and z axes. The rear plane is marked x = y\/2 or y = 2x. The front plane is marked x = y\/2 + 1 or y = 2x minus 2.\" width=\"894\" height=\"435\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/04142142\/9ac91afe22764bb4840cdcc0310bae65f7605c69.webp 894w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/04142142\/9ac91afe22764bb4840cdcc0310bae65f7605c69-300x146.webp 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/04142142\/9ac91afe22764bb4840cdcc0310bae65f7605c69-768x374.webp 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/04142142\/9ac91afe22764bb4840cdcc0310bae65f7605c69-65x32.webp 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/04142142\/9ac91afe22764bb4840cdcc0310bae65f7605c69-225x109.webp 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/68\/2025\/10\/04142142\/9ac91afe22764bb4840cdcc0310bae65f7605c69-350x170.webp 350w\" sizes=\"auto, (max-width: 894px) 100vw, 894px\" \/><figcaption id=\"caption-attachment-1983\" class=\"wp-caption-text\">Figure 9. The region [latex]G[\/latex] in [latex]uvw[\/latex]-space is transformed to region D in [latex]xyz[\/latex]-space.<\/figcaption><\/figure>\n<\/div>\n<div><\/div>\n<\/div>\n<p id=\"fs-id1167793607338\">Now we can calculate the Jacobian for the transformation:<\/p>\n<p style=\"text-align: center;\">[latex]\\large{J(u,v,w)=\\begin{vmatrix}\\frac{\\partial{x}}{\\partial{u}}&\\frac{\\partial{x}}{\\partial{v}}&\\frac{\\partial{x}}{\\partial{w}} \\\\ \\frac{\\partial{y}}{\\partial{u}}&\\frac{\\partial{y}}{\\partial{v}}&\\frac{\\partial{y}}{\\partial{w}} \\\\ \\frac{\\partial{z}}{\\partial{u}}&\\frac{\\partial{z}}{\\partial{v}}&\\frac{\\partial{z}}{\\partial{w}}\\end{vmatrix}=\\begin{vmatrix}1&1&0 \\\\ 0&2&0 \\\\ 0&0&3\\end{vmatrix}=6}[\/latex].<\/p>\n<p id=\"fs-id1167794296460\">The function to be integrated becomes<\/p>\n<p style=\"text-align: center;\">[latex]\\large{f(x,y,z)=x+\\frac{z}3=u+v+\\frac{3w}3=u+v+w}[\/latex].<\/p>\n<p id=\"fs-id1167793420356\">We are now ready to put everything together and complete the problem.<\/p>\n<p>[latex]\\hspace{3cm}\\large{\\begin{align}    &\\hspace{1cm}\\displaystyle\\int_0^3\\displaystyle\\int_0^4\\displaystyle\\int_{y\/2}^{(y\/2)+1}\\left(x+\\frac{z}3\\right)dx \\ dy \\ dz \\\\    &=\\displaystyle\\int_0^1\\displaystyle\\int_0^2\\displaystyle\\int_0^1(u+v+w)|J(u,v,w)|du \\ dv \\ dw=\\displaystyle\\int_0^1\\displaystyle\\int_0^2\\displaystyle\\int_0^1(u+v+w)|6|du \\ dv \\ dw \\\\    &=6\\displaystyle\\int_0^1\\displaystyle\\int_0^2\\displaystyle\\int_0^1(u+v+w)du \\ dv \\ dw=6\\displaystyle\\int_0^1\\displaystyle\\int_0^2\\left[\\frac{u^2}2+vu+wu\\right]_0^1dv \\ dw \\\\    &=6\\displaystyle\\int_0^1\\displaystyle\\int_0^2\\left(\\frac12+v+w\\right)dv \\ dw=6\\displaystyle\\int_0^1\\left[\\frac12v+\\frac{v^2}2+wv\\right]_0^2dw \\\\    &=6\\displaystyle\\int_0^1(3+2w)dw=6\\left[3w+w^2\\right]_0^1=24    \\end{align}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":25,"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-1305","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\/1305","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":8,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1305\/revisions"}],"predecessor-version":[{"id":1988,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1305\/revisions\/1988"}],"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\/1305\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=1305"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=1305"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=1305"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=1305"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}