{"id":702,"date":"2025-10-09T15:33:03","date_gmt":"2025-10-09T15:33:03","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=702"},"modified":"2025-10-31T16:19:34","modified_gmt":"2025-10-31T16:19:34","slug":"triple-integration-and-applications-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/triple-integration-and-applications-cheat-sheet\/","title":{"raw":"Triple Integration and Applications: Cheat Sheet","rendered":"Triple Integration and Applications: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<strong>Triple Integrals<\/strong>\r\n<ul>\r\n \t<li>To compute a triple integral we use Fubini\u2019s theorem, which states that if [latex]f(x,y,z)[\/latex]\u00a0is continuous on a rectangular box<\/li>\r\n \t<li>To compute the volume of a general solid bounded region [latex]E[\/latex]\u00a0we use the triple integral<\/li>\r\n \t<li>Interchanging the order of the iterated integrals does not change the answer. As a matter of fact, interchanging the order of integration can help simplify the computation.<\/li>\r\n \t<li>To compute the average value of a function over a general three-dimensional region, we use<\/li>\r\n<\/ul>\r\n<strong>Triple Integrals in Alternative Coordinate Systems<\/strong>\r\n<ul>\r\n \t<li>To evaluate a triple integral in cylindrical coordinates, use the iterated integral<\/li>\r\n \t<li>To evaluate a triple integral in spherical coordinates, use the iterated integral<\/li>\r\n<\/ul>\r\n<strong>Center of Mass and Moments of Inertia<\/strong>\r\n<ul>\r\n \t<li>Finding the mass, center of mass, moments, and moments of inertia in double integrals:\r\n<ul>\r\n \t<li>For a lamina\u00a0[latex]R[\/latex]\u00a0with a density function\u00a0[latex]\\rho(x,y)[\/latex] at any point\u00a0[latex](x,y)[\/latex]\u00a0in the plane, the mass is [latex]m=\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA[\/latex]<\/li>\r\n \t<li>The moments about the\u00a0[latex]x[\/latex]-axis and [latex]y[\/latex]-axis are [latex]M_{x}=\\underset{R}{\\displaystyle\\iint} y\\rho(x,y) dA[\/latex] and\u00a0[latex]M_{y}=\\underset{R}{\\displaystyle\\iint} x\\rho(x,y) dA[\/latex]<\/li>\r\n \t<li>The center of mass is given by\u00a0[latex]\\overline{x}=\\frac{M_{y}}{m}[\/latex],\u00a0[latex]\\overline{y}=\\frac{M_{x}}{m}[\/latex]<\/li>\r\n \t<li>The center of mass becomes the centroid of the plane when the density is constant.<\/li>\r\n \t<li>The moments of inertia about the [latex]x[\/latex]-axis,\u00a0[latex]y[\/latex]-axis, and the origin are\u00a0[latex]I_{x}=\\displaystyle\\iint_{R} y^{2}\\rho(x,y) dA[\/latex],\u00a0[latex]I_{y}=\\underset{R}{\\displaystyle\\iint} x^{2}\\rho(x,y) dA[\/latex], and\u00a0[latex]I_{0}=I_{x}+I_{y}=\\underset{R}{\\displaystyle\\iint} (x^{2}+y^{2})\\rho(x,y) dA[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Finding the mass, center of mass, moments, and moments of inertia in triple integrals:\r\n<ul>\r\n \t<li>For a solid object\u00a0[latex]Q[\/latex]\u00a0with a density function\u00a0[latex]\\rho(x,y,z)[\/latex] at any point\u00a0[latex](x,y,z)[\/latex]\u00a0in space, the mass is [latex]m=\\displaystyle\\iiint_{Q} \\rho(x,y,z) dV[\/latex]<\/li>\r\n \t<li>The moments about the\u00a0[latex]xy[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the [latex]yz[\/latex]-plane are\u00a0[latex]M_{xy}=\\displaystyle\\iiint_{Q} z\\rho(x,y,z) dV[\/latex],\u00a0[latex]M_{xz}=\\underset{Q}{\\displaystyle\\iiint} y\\rho(x,y,z) dV[\/latex],\u00a0[latex]M_{yz}=\\underset{Q}{\\displaystyle\\iiint} x\\rho(x,y,z) dV[\/latex]<\/li>\r\n \t<li>The center of mass is given by\u00a0[latex]\\overline{x}=\\frac{M_{yz}}{m}[\/latex],\u00a0[latex]\\overline{x}=\\frac{M_{xz}}{m}[\/latex],\u00a0[latex]\\overline{x}=\\frac{M_{xy}}{m}[\/latex]<\/li>\r\n \t<li>The center of mass becomes the centroid of the solid when the density is constant.<\/li>\r\n \t<li>The moments of inertia about the\u00a0[latex]yz[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the [latex]xy[\/latex]-plane are\u00a0[latex]I_{x}=\\underset{Q}{\\displaystyle\\iiint} (y^{2}+z^{2})\\rho(x,y,z) dV[\/latex],\u00a0[latex]I_{y}=\\underset{Q}{\\displaystyle\\iiint} (x^{2}+z^{2})\\rho(x,y,z) dV[\/latex],\u00a0[latex]I_{z}=\\underset{Q}{\\displaystyle\\iiint} (x^{2}+y^{2})\\rho(x,y,z) dV[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>Change of Variables<\/strong>\r\n<ul>\r\n \t<li>A transformation\u00a0[latex]T[\/latex]\u00a0is a function that transforms a region\u00a0[latex]G[\/latex]\u00a0in one plane (space) into a region\u00a0[latex]R[\/latex]\u00a0in another plane (space) by a change of variables.<\/li>\r\n \t<li>A transformation\u00a0[latex]T:G\\rightarrow{R}[\/latex] defined as\u00a0[latex]T(u,v)=(x,y)[\/latex] [latex](\\text{or }T(u,v,w)=(x,y,z))([\/latex]\u00a0is said to be a one-to-one transformation if no two points map to the same image point.<\/li>\r\n \t<li>If [latex]f[\/latex]\u00a0is continuous on\u00a0[latex]R[\/latex], then\u00a0[latex]\\underset{R}{\\displaystyle\\iint} f(x,y) dA =\\underset{S}{\\displaystyle\\iint} f\\left(g(u,v),h(u,v)\\right)\\left\\arrowvert\\frac{\\partial(x,y)}{\\partial(u,v)}\\right\\arrowvert du dv[\/latex]<\/li>\r\n \t<li>If [latex]F[\/latex]\u00a0is continuous on\u00a0[latex]R[\/latex], then\u00a0[latex]R[\/latex], then\u00a0[latex]\\underset{R}{\\displaystyle\\iiint} F(x,y,z) dV =\\underset{G}{\\displaystyle\\iiint} F\\left(g(u,v,w),h(u,v,w),k(u,v,w)\\right)\\left\\arrowvert\\frac{\\partial(x,y,z)}{\\partial(u,v,w)}\\right\\arrowvert du dv dw = \\underset{G}{\\displaystyle\\iiint}H(u,v,w)|J(u,v,w)| du dv dw[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<ul>\r\n \t<li><strong>Triple integral\r\n<\/strong>[latex]\\underset{l,m,n\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{l}\\displaystyle\\sum_{j=1}^{m}\\displaystyle\\sum_{k=1}^{n} f(x_{ijk}^{\\ast},y_{ijk}^{\\ast},z_{ijk}^{\\ast}) \\Delta{x}\\Delta{y}\\Delta{z}=\\underset{B}{\\displaystyle\\iiint} f(x,y,z)dV[\/latex]<\/li>\r\n \t<li><strong>Triple integral in cylindrical coordinates\r\n<\/strong>[latex]\\underset{B}{\\displaystyle\\iiint} g(x,y,z)dV=\\underset{B}{\\displaystyle\\iiint} g(r\\cos\\theta,r\\sin\\theta,z)r dr d\\theta dz=\\underset{B}{\\displaystyle\\iiint} f(r,\\theta,z)r dr d\\theta dz=[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Triple integral in spherical coordinates<\/strong>\r\n[latex]\\underset{B}{\\displaystyle\\iiint} f(\\rho,\\theta,\\varphi)\\rho^{2}\\sin\\varphi{d}\\rho{d}\\varphi{d}\\theta=\\displaystyle\\int_{\\varphi=\\gamma}^{\\varphi=\\psi}\\displaystyle\\int_{\\theta=\\alpha}^{\\theta=\\beta}\\displaystyle\\int_{\\rho=a}^{\\rho=b} \\rho^{2}\\sin\\varphi{d}\\rho{d}\\varphi{d}\\theta[\/latex]<\/li>\r\n \t<li><strong>Mass of a lamina\r\n<\/strong>[latex]m=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}m_{ij}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}\\rho(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}=\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA[\/latex]<\/li>\r\n \t<li><strong>Moment about the [latex]x[\/latex]-axis\r\n<\/strong>[latex]M_{x}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(y_{ij}^{\\ast})m_{ij}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(y_{ij}^{\\ast})\\rho(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}=\\underset{R}{\\displaystyle\\iint} y\\rho(x,y) dA[\/latex]<\/li>\r\n \t<li><strong>Moment about the [latex]y[\/latex]-axis\r\n<\/strong>[latex]M_{y}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(x_{ij}^{\\ast})m_{ij}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(x_{ij}^{\\ast})\\rho(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}=\\underset{R}{\\displaystyle\\iint} x\\rho(x,y) dA[\/latex]<\/li>\r\n \t<li><strong>Center of mass of a lamina\r\n<\/strong>[latex]\\overline{x}=\\frac{M_{y}}{m}=\\dfrac{\\underset{R}{\\displaystyle\\iint} x\\rho(x,y) dA}{\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA}[\/latex] and [latex]\\overline{y}=\\frac{M_{x}}{m}=\\dfrac{\\underset{R}{\\displaystyle\\iint} y\\rho(x,y) dA}{\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>Jacobian<\/dt>\r\n \t<dd>the Jacobian [latex]J(u ,v)[\/latex] in two variables is a [latex]2{\\times}2[\/latex] determinant:<\/dd>\r\n \t<dd>[latex]J(u,v) = \\begin{vmatrix}\\frac{dx}{du} &amp; \\frac{dy}{du}\\\\\\frac{dx}{dv} &amp; \\frac{dy}{dv}\\end{vmatrix}[\/latex]<\/dd>\r\n \t<dd>the Jacobian [latex]J(u ,v, w)[\/latex]\u00a0in three variables is a [latex]3{\\times}3[\/latex]\u00a0determinant:<\/dd>\r\n \t<dd>[latex]J(u,v,w)=\\begin{vmatrix}\\frac{dx}{du} &amp; \\frac{dy}{du} &amp; \\frac{dz}{du}\\\\\\frac{dx}{dv} &amp; \\frac{dy}{dv} &amp; \\frac{dz}{dv}\\\\\\frac{dx}{dw} &amp; \\frac{dy}{dw} &amp; \\frac{dz}{dw}\\end{vmatrix}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>one-to-one transformation<\/dt>\r\n \t<dd>a transformation [latex]T : G {\\rightarrow} R[\/latex]\u00a0defined as [latex]T(u, v) = (x, y)[\/latex]\u00a0is said to be one-to-one if no two points map to the same image point<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>planar transformation<\/dt>\r\n \t<dd>a function [latex]T[\/latex]\u00a0that transforms a region [latex]G[\/latex]\u00a0in one plane into a region [latex]R[\/latex]\u00a0in another plane by a change of variables<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>radius of gyration<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">the distance from an object\u2019s center of mass to its axis of rotation<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>transformation<\/dt>\r\n \t<dd>a function that transforms a region [latex]G[\/latex] in one plane into a region [latex]R[\/latex]\u00a0in another plane by a change of variables<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>triple integral<\/dt>\r\n \t<dd>the triple integral of a continuous function [latex]f(x, y, z)[\/latex]over a rectangular solid box [latex]\\bf{B}[\/latex] is the limit of a Riemann sum for a function of three variables, if this limit exists<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>triple integral in cylindrical coordinates<\/dt>\r\n \t<dd>the limit of a triple Riemann sum, provided the following limit exists:[latex]{\\displaystyle\\lim_{l,m,n\\to\\infty}{\\sum_{i=1}^{l}}{\\displaystyle\\sum_{j=1}^{m}}{\\displaystyle\\sum_{k=1}^{n}f({r^{*}}_{i,j,k}, {{\\theta}^{*}}_{i,j,k}, {{z}^{*}}_{i,j,k}){r^{*}}_{i,j,k}{\\Delta}r{\\Delta}{\\theta}{\\Delta}{z}}}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>triple integral in spherical coordinates<\/dt>\r\n \t<dd>the limit of a triple Riemann sum, provided the following limit exists:\u00a0[latex]{\\displaystyle\\lim_{l,m,n\\to\\infty}{\\displaystyle\\sum_{i=1}^{l}}{\\displaystyle\\sum_{j=1}^{m}}{\\displaystyle\\sum_{k=1}^{n}f({{\\rho}^{*}}_{i,j,k}, {{\\theta}^{*}}_{i,j,k}, {{\\varphi}^{*}}_{i,j,k})({{\\rho}^{*}}_{i,j,k})^{2}\\sin{\\varphi}{\\Delta}{\\rho}{\\Delta}{\\theta}{\\Delta}{\\varphi}}}[\/latex]<\/dd>\r\n<\/dl>","rendered":"<h2>Essential Concepts<\/h2>\n<p><strong>Triple Integrals<\/strong><\/p>\n<ul>\n<li>To compute a triple integral we use Fubini\u2019s theorem, which states that if [latex]f(x,y,z)[\/latex]\u00a0is continuous on a rectangular box<\/li>\n<li>To compute the volume of a general solid bounded region [latex]E[\/latex]\u00a0we use the triple integral<\/li>\n<li>Interchanging the order of the iterated integrals does not change the answer. As a matter of fact, interchanging the order of integration can help simplify the computation.<\/li>\n<li>To compute the average value of a function over a general three-dimensional region, we use<\/li>\n<\/ul>\n<p><strong>Triple Integrals in Alternative Coordinate Systems<\/strong><\/p>\n<ul>\n<li>To evaluate a triple integral in cylindrical coordinates, use the iterated integral<\/li>\n<li>To evaluate a triple integral in spherical coordinates, use the iterated integral<\/li>\n<\/ul>\n<p><strong>Center of Mass and Moments of Inertia<\/strong><\/p>\n<ul>\n<li>Finding the mass, center of mass, moments, and moments of inertia in double integrals:\n<ul>\n<li>For a lamina\u00a0[latex]R[\/latex]\u00a0with a density function\u00a0[latex]\\rho(x,y)[\/latex] at any point\u00a0[latex](x,y)[\/latex]\u00a0in the plane, the mass is [latex]m=\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA[\/latex]<\/li>\n<li>The moments about the\u00a0[latex]x[\/latex]-axis and [latex]y[\/latex]-axis are [latex]M_{x}=\\underset{R}{\\displaystyle\\iint} y\\rho(x,y) dA[\/latex] and\u00a0[latex]M_{y}=\\underset{R}{\\displaystyle\\iint} x\\rho(x,y) dA[\/latex]<\/li>\n<li>The center of mass is given by\u00a0[latex]\\overline{x}=\\frac{M_{y}}{m}[\/latex],\u00a0[latex]\\overline{y}=\\frac{M_{x}}{m}[\/latex]<\/li>\n<li>The center of mass becomes the centroid of the plane when the density is constant.<\/li>\n<li>The moments of inertia about the [latex]x[\/latex]-axis,\u00a0[latex]y[\/latex]-axis, and the origin are\u00a0[latex]I_{x}=\\displaystyle\\iint_{R} y^{2}\\rho(x,y) dA[\/latex],\u00a0[latex]I_{y}=\\underset{R}{\\displaystyle\\iint} x^{2}\\rho(x,y) dA[\/latex], and\u00a0[latex]I_{0}=I_{x}+I_{y}=\\underset{R}{\\displaystyle\\iint} (x^{2}+y^{2})\\rho(x,y) dA[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Finding the mass, center of mass, moments, and moments of inertia in triple integrals:\n<ul>\n<li>For a solid object\u00a0[latex]Q[\/latex]\u00a0with a density function\u00a0[latex]\\rho(x,y,z)[\/latex] at any point\u00a0[latex](x,y,z)[\/latex]\u00a0in space, the mass is [latex]m=\\displaystyle\\iiint_{Q} \\rho(x,y,z) dV[\/latex]<\/li>\n<li>The moments about the\u00a0[latex]xy[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the [latex]yz[\/latex]-plane are\u00a0[latex]M_{xy}=\\displaystyle\\iiint_{Q} z\\rho(x,y,z) dV[\/latex],\u00a0[latex]M_{xz}=\\underset{Q}{\\displaystyle\\iiint} y\\rho(x,y,z) dV[\/latex],\u00a0[latex]M_{yz}=\\underset{Q}{\\displaystyle\\iiint} x\\rho(x,y,z) dV[\/latex]<\/li>\n<li>The center of mass is given by\u00a0[latex]\\overline{x}=\\frac{M_{yz}}{m}[\/latex],\u00a0[latex]\\overline{x}=\\frac{M_{xz}}{m}[\/latex],\u00a0[latex]\\overline{x}=\\frac{M_{xy}}{m}[\/latex]<\/li>\n<li>The center of mass becomes the centroid of the solid when the density is constant.<\/li>\n<li>The moments of inertia about the\u00a0[latex]yz[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the [latex]xy[\/latex]-plane are\u00a0[latex]I_{x}=\\underset{Q}{\\displaystyle\\iiint} (y^{2}+z^{2})\\rho(x,y,z) dV[\/latex],\u00a0[latex]I_{y}=\\underset{Q}{\\displaystyle\\iiint} (x^{2}+z^{2})\\rho(x,y,z) dV[\/latex],\u00a0[latex]I_{z}=\\underset{Q}{\\displaystyle\\iiint} (x^{2}+y^{2})\\rho(x,y,z) dV[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>Change of Variables<\/strong><\/p>\n<ul>\n<li>A transformation\u00a0[latex]T[\/latex]\u00a0is a function that transforms a region\u00a0[latex]G[\/latex]\u00a0in one plane (space) into a region\u00a0[latex]R[\/latex]\u00a0in another plane (space) by a change of variables.<\/li>\n<li>A transformation\u00a0[latex]T:G\\rightarrow{R}[\/latex] defined as\u00a0[latex]T(u,v)=(x,y)[\/latex] [latex](\\text{or }T(u,v,w)=(x,y,z))([\/latex]\u00a0is said to be a one-to-one transformation if no two points map to the same image point.<\/li>\n<li>If [latex]f[\/latex]\u00a0is continuous on\u00a0[latex]R[\/latex], then\u00a0[latex]\\underset{R}{\\displaystyle\\iint} f(x,y) dA =\\underset{S}{\\displaystyle\\iint} f\\left(g(u,v),h(u,v)\\right)\\left\\arrowvert\\frac{\\partial(x,y)}{\\partial(u,v)}\\right\\arrowvert du dv[\/latex]<\/li>\n<li>If [latex]F[\/latex]\u00a0is continuous on\u00a0[latex]R[\/latex], then\u00a0[latex]R[\/latex], then\u00a0[latex]\\underset{R}{\\displaystyle\\iiint} F(x,y,z) dV =\\underset{G}{\\displaystyle\\iiint} F\\left(g(u,v,w),h(u,v,w),k(u,v,w)\\right)\\left\\arrowvert\\frac{\\partial(x,y,z)}{\\partial(u,v,w)}\\right\\arrowvert du dv dw = \\underset{G}{\\displaystyle\\iiint}H(u,v,w)|J(u,v,w)| du dv dw[\/latex]<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul>\n<li><strong>Triple integral<br \/>\n<\/strong>[latex]\\underset{l,m,n\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{l}\\displaystyle\\sum_{j=1}^{m}\\displaystyle\\sum_{k=1}^{n} f(x_{ijk}^{\\ast},y_{ijk}^{\\ast},z_{ijk}^{\\ast}) \\Delta{x}\\Delta{y}\\Delta{z}=\\underset{B}{\\displaystyle\\iiint} f(x,y,z)dV[\/latex]<\/li>\n<li><strong>Triple integral in cylindrical coordinates<br \/>\n<\/strong>[latex]\\underset{B}{\\displaystyle\\iiint} g(x,y,z)dV=\\underset{B}{\\displaystyle\\iiint} g(r\\cos\\theta,r\\sin\\theta,z)r dr d\\theta dz=\\underset{B}{\\displaystyle\\iiint} f(r,\\theta,z)r dr d\\theta dz=[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Triple integral in spherical coordinates<\/strong><br \/>\n[latex]\\underset{B}{\\displaystyle\\iiint} f(\\rho,\\theta,\\varphi)\\rho^{2}\\sin\\varphi{d}\\rho{d}\\varphi{d}\\theta=\\displaystyle\\int_{\\varphi=\\gamma}^{\\varphi=\\psi}\\displaystyle\\int_{\\theta=\\alpha}^{\\theta=\\beta}\\displaystyle\\int_{\\rho=a}^{\\rho=b} \\rho^{2}\\sin\\varphi{d}\\rho{d}\\varphi{d}\\theta[\/latex]<\/li>\n<li><strong>Mass of a lamina<br \/>\n<\/strong>[latex]m=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}m_{ij}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}\\rho(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}=\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA[\/latex]<\/li>\n<li><strong>Moment about the [latex]x[\/latex]-axis<br \/>\n<\/strong>[latex]M_{x}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(y_{ij}^{\\ast})m_{ij}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(y_{ij}^{\\ast})\\rho(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}=\\underset{R}{\\displaystyle\\iint} y\\rho(x,y) dA[\/latex]<\/li>\n<li><strong>Moment about the [latex]y[\/latex]-axis<br \/>\n<\/strong>[latex]M_{y}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(x_{ij}^{\\ast})m_{ij}=\\underset{k,l\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{k}\\displaystyle\\sum_{j=1}^{l}(x_{ij}^{\\ast})\\rho(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}=\\underset{R}{\\displaystyle\\iint} x\\rho(x,y) dA[\/latex]<\/li>\n<li><strong>Center of mass of a lamina<br \/>\n<\/strong>[latex]\\overline{x}=\\frac{M_{y}}{m}=\\dfrac{\\underset{R}{\\displaystyle\\iint} x\\rho(x,y) dA}{\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA}[\/latex] and [latex]\\overline{y}=\\frac{M_{x}}{m}=\\dfrac{\\underset{R}{\\displaystyle\\iint} y\\rho(x,y) dA}{\\underset{R}{\\displaystyle\\iint} \\rho(x,y) dA}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>Jacobian<\/dt>\n<dd>the Jacobian [latex]J(u ,v)[\/latex] in two variables is a [latex]2{\\times}2[\/latex] determinant:<\/dd>\n<dd>[latex]J(u,v) = \\begin{vmatrix}\\frac{dx}{du} & \\frac{dy}{du}\\\\\\frac{dx}{dv} & \\frac{dy}{dv}\\end{vmatrix}[\/latex]<\/dd>\n<dd>the Jacobian [latex]J(u ,v, w)[\/latex]\u00a0in three variables is a [latex]3{\\times}3[\/latex]\u00a0determinant:<\/dd>\n<dd>[latex]J(u,v,w)=\\begin{vmatrix}\\frac{dx}{du} & \\frac{dy}{du} & \\frac{dz}{du}\\\\\\frac{dx}{dv} & \\frac{dy}{dv} & \\frac{dz}{dv}\\\\\\frac{dx}{dw} & \\frac{dy}{dw} & \\frac{dz}{dw}\\end{vmatrix}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>one-to-one transformation<\/dt>\n<dd>a transformation [latex]T : G {\\rightarrow} R[\/latex]\u00a0defined as [latex]T(u, v) = (x, y)[\/latex]\u00a0is said to be one-to-one if no two points map to the same image point<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>planar transformation<\/dt>\n<dd>a function [latex]T[\/latex]\u00a0that transforms a region [latex]G[\/latex]\u00a0in one plane into a region [latex]R[\/latex]\u00a0in another plane by a change of variables<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>radius of gyration<\/dt>\n<dd><span style=\"font-size: 1em;\">the distance from an object\u2019s center of mass to its axis of rotation<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>transformation<\/dt>\n<dd>a function that transforms a region [latex]G[\/latex] in one plane into a region [latex]R[\/latex]\u00a0in another plane by a change of variables<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>triple integral<\/dt>\n<dd>the triple integral of a continuous function [latex]f(x, y, z)[\/latex]over a rectangular solid box [latex]\\bf{B}[\/latex] is the limit of a Riemann sum for a function of three variables, if this limit exists<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>triple integral in cylindrical coordinates<\/dt>\n<dd>the limit of a triple Riemann sum, provided the following limit exists:[latex]{\\displaystyle\\lim_{l,m,n\\to\\infty}{\\sum_{i=1}^{l}}{\\displaystyle\\sum_{j=1}^{m}}{\\displaystyle\\sum_{k=1}^{n}f({r^{*}}_{i,j,k}, {{\\theta}^{*}}_{i,j,k}, {{z}^{*}}_{i,j,k}){r^{*}}_{i,j,k}{\\Delta}r{\\Delta}{\\theta}{\\Delta}{z}}}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>triple integral in spherical coordinates<\/dt>\n<dd>the limit of a triple Riemann sum, provided the following limit exists:\u00a0[latex]{\\displaystyle\\lim_{l,m,n\\to\\infty}{\\displaystyle\\sum_{i=1}^{l}}{\\displaystyle\\sum_{j=1}^{m}}{\\displaystyle\\sum_{k=1}^{n}f({{\\rho}^{*}}_{i,j,k}, {{\\theta}^{*}}_{i,j,k}, {{\\varphi}^{*}}_{i,j,k})({{\\rho}^{*}}_{i,j,k})^{2}\\sin{\\varphi}{\\Delta}{\\rho}{\\Delta}{\\theta}{\\Delta}{\\varphi}}}[\/latex]<\/dd>\n<\/dl>\n","protected":false},"author":15,"menu_order":1,"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-702","chapter","type-chapter","status-publish","hentry"],"part":434,"module-header":"cheat_sheet","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\/702","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\/702\/revisions"}],"predecessor-version":[{"id":1933,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/702\/revisions\/1933"}],"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\/702\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=702"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=702"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=702"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=702"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}