{"id":1072,"date":"2021-11-01T19:18:25","date_gmt":"2021-11-01T19:18:25","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=1072"},"modified":"2022-11-01T04:39:11","modified_gmt":"2022-11-01T04:39:11","slug":"center-of-mass-and-moments-of-inertia-in-three-dimensions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/center-of-mass-and-moments-of-inertia-in-three-dimensions\/","title":{"raw":"Center of Mass and Moments of Inertia in Three Dimensions","rendered":"Center of Mass and Moments of Inertia in Three Dimensions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3 style=\"text-align: center;\">Learning Objectives<\/h3>\r\n<ul class=\"os-abstract\">\r\n \t<li><span class=\"os-abstract-content\">Use triple integrals to locate the center of mass of a three-dimensional object.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1167793294160\">All the expressions of double integrals discussed so far can be modified to become triple integrals.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1167793294168\">If we have a solid object [latex]Q[\/latex]\u00a0with a density function [latex]{\\rho}{({x},{y},{z})}[\/latex] at any point[latex](x, y, z)[\/latex] in space, then its mass is<\/p>\r\n<p style=\"text-align: center;\">[latex]{m} = {\\underset{Q}{\\displaystyle\\iiint}}{\\rho}{({x},{y},{z})}{dV}.[\/latex]<\/p>\r\n<p id=\"fs-id1167793252305\">Its moments about the [latex]xy[\/latex]-plane, the [latex]xz[\/latex]-plane, and the [latex]yz[\/latex]-plane are<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\r\n{{M}_{xy}} = &amp; {\\underset{Q}{\\displaystyle\\iiint}}{z}{\\rho}{({x},{y},{z})}{dV}, {{M}_{xz}} = {\\underset{Q}{\\displaystyle\\iiint}}{y}{\\rho}{({x},{y},{z})}{dV}, \\\\\r\n{{M}_{yz}} = &amp; {\\underset{Q}{\\displaystyle\\iiint}}{x}{\\rho}{({x},{y},{z})}{dV}.\r\n\\end{aligned}[\/latex]<\/p>\r\n<p id=\"fs-id1167793499101\">If the center of mass of the object is the point [latex]{\\left ( {\\overline{x}},{\\overline{y}},{\\overline{z}} \\right )},[\/latex] then<\/p>\r\n<p style=\"text-align: center;\">[latex]\\large{{\\overline{x}} = {\\dfrac{{M}_{yz}}{m}}, \\ {\\overline{y}} = {\\dfrac{{M}_{xz}}{m}}, \\ {\\overline{z}} = {\\dfrac{{M}_{xy}}{m}}}.[\/latex]<\/p>\r\n<p id=\"fs-id1167794181252\">Also, if the solid object is homogeneous (with constant density), then the center of mass becomes the centroid of the solid. Finally, the moments of inertia about the [latex]yz[\/latex]-plane, the [latex]xz[\/latex]-plane,\u00a0<span id=\"MathJax-Element-184-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; overflow: initial; display: inline-table; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;\/mi&gt;&lt;mi&gt;z&lt;\/mi&gt;&lt;mtext&gt;-plane,&lt;\/mtext&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;\/mi&gt;&lt;mi&gt;z&lt;\/mi&gt;&lt;mtext&gt;-plane,&lt;\/mtext&gt;&lt;\/mrow&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt;\"><\/span>and the [latex]xy[\/latex]-plane are<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\r\n{{I}_{x}} = &amp; {\\underset{Q}{\\displaystyle\\iiint}}{({{y}^{2}} + {{z}^{2}})}{\\rho}{({x},{y},{z})}{dV}, \\\\\r\n{{I}_{y}} = &amp;{\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{z}^{2}})}{\\rho}{({x},{y},{z})}{dV}, \\\\\r\n{{I}_{z}} = &amp; {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{y}^{2}})}{\\rho}{({x},{y},{z})}{dV}.\r\n\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: finding the mass of a solid<\/h3>\r\nSuppose that [latex]Q[\/latex] is a solid region bounded by [latex]x+2y+3z=6[\/latex] and the coordinate planes and has density [latex]{\\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[\/latex]. Find the total mass.\r\n\r\n[reveal-answer q=\"875204756\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"875204756\"]\r\n<p id=\"fs-id1167793589656\">The region [latex]Q[\/latex] is a tetrahedron (Figure 1) meeting the axes at the points [latex](6, 0, 0)[\/latex], [latex](0, 3, 0)[\/latex], and [latex](0, 0, 2)[\/latex]. To find the limits of integration, let [latex]z=0[\/latex] in the slanted plane [latex]{z} = {\\frac{1}{3}}{({6} - {x} - {{2}{y}})}[\/latex]. Then for [latex]x[\/latex] and [latex]y[\/latex] find the projection of [latex]Q[\/latex] onto the [latex]xy[\/latex]-plane, which is bounded by the axes and the line [latex]x+2y=6[\/latex]. Hence the mass is<\/p>\r\n\r\n<center>\r\n[latex]\r\n{m} = {\\underset{Q}{\\displaystyle\\iiint}}{\\rho}{({x},{y},{z})}{dV} = \\displaystyle\\int^{{x} = {6}}_{{x} = {0}} \\ \\displaystyle\\int^{y = \\frac{1}{2}({6} - x)}_{y = 0} \\ \\displaystyle\\int^{z = \\frac{1}{3}(6 - x - 2y)}_{z = 0} \\ x^{2}yz \\ dz \\ dy \\ dx = \\frac{108}{35} \\ \\approx \\ 3.086.\r\n[\/latex]<\/center>[caption id=\"attachment_1414\" align=\"aligncenter\" width=\"268\"]<img class=\"size-full wp-image-1414\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/26174918\/5-6-7.jpeg\" alt=\"In x y z space, the solid Q is shown with corners (0, 0, 0), (0, 0, 2), (0, 3, 0), and (6, 0, 0). Alternatively, you could consider the solid as being bounded by the x y, x z, and y z planes and the plane x + 2y + 3z = 6, forming an irregular tetrahedron.\" width=\"268\" height=\"309\" \/> Figure 1. Finding the mass of a three-dimensional solid [latex]Q[\/latex].[\/caption][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nConsider the same region [latex]Q[\/latex] (Figure 1), and use the density function [latex]{\\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[\/latex].\u00a0<span style=\"font-size: 1rem; text-align: initial;\">Find the mass.<\/span>\r\n\r\n[reveal-answer q=\"685720429\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"685720429\"]\r\n\r\n[latex]\\frac{54}{35}=1.543[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: finding the center of Mass of a solid<\/h3>\r\nSuppose [latex]Q[\/latex] is a solid region bounded by the plane [latex]x+2y+3z=6[\/latex] and the coordinate planes with density [latex]{\\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[\/latex] (see\u00a0Figure 1). Find the center of mass using decimal approximation. Use the mass found in\u00a0Example \"Finding the Mass of a Solid\".\r\n\r\n[reveal-answer q=\"723096348\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"723096348\"]\r\n<p id=\"fs-id1167793777311\">We have used this tetrahedron before and know the limits of integration, so we can proceed to the computations right away. First, we need to find the moments about the\u00a0[latex]xy[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the\u00a0[latex]yz[\/latex]-plane:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\r\n{{M}_{xy}} = &amp;{\\underset{Q}{\\displaystyle\\iiint}}{z}{\\rho}{({x},{y},{z})}{dV} = {\\displaystyle\\int^{{x} = {6}}_{{x} = {0}}} \\ {\\displaystyle\\int^{{y} = {1\/2}{({6} - {x})}}_{{y} = {0}}} \\ {\\displaystyle\\int^{{z} = {1\/3}{({6} - {x} - {2y})}}_{{z} = {0}}} \\ {{x}^{2}}{{yz}^{2}}{dz} \\ {dy} \\ {dx} = {\\frac{54}{35}} \\ {\\approx} \\ {1.543}, \\\\\r\n{{M}_{xz}} = &amp; {\\underset{Q}{\\displaystyle\\iiint}}{y}{\\rho}{({x},{y},{z})}{dV} = {\\displaystyle\\int^{{x} = {6}}_{{x} = {0}}} \\ {\\displaystyle\\int^{{y} = {1\/2}{({6} - {x})}}_{{y} = {0}}} \\ {\\displaystyle\\int^{{z} = {1\/3}{({6} - {x} - {2y})}}_{{z} = {0}}} \\ {{x}^{2}}{{y}^{2}}{z} \\ {dz} \\ {dy} \\ {dx} = {\\frac{81}{35}} \\ {\\approx} \\ {2.314}, \\\\\r\n{{M}_{yz}} = &amp; {\\underset{Q}{\\displaystyle\\iiint}}{x}{\\rho}{({x},{y},{z})}{dV} = {\\displaystyle\\int^{{x} = {6}}_{{x} = {0}}} \\ {\\displaystyle\\int^{{y} = {1\/2}{({6} - {x})}}_{{y} = {0}}} \\ {\\displaystyle\\int^{{z} = {1\/3}{({6} - {x} - {2y})}}_{{z} = {0}}} \\ {{x}^{3}}{yz} \\ {dz} \\ {dy} \\ {dx} = {\\frac{243}{35}} \\ {\\approx} \\ {6.943}.\r\n\\end{aligned}[\/latex]<\/p>\r\nHence the center of mass is\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\r\n{\\overline{x}} = &amp; {\\frac{{M}_{yz}}{m}}, \\ {\\overline{y}} = {\\frac{{M}_{xz}}{m}}, \\ {\\overline{z}} = {\\frac{{M}_{xy}}{m}}, \\\\\r\n{\\overline{x}} = &amp; {\\frac{{M}_{yz}}{m}} = {\\frac{243\/35}{108\/35}} = {\\frac{243}{108}} = {2.25}, \\\\\r\n{\\overline{y}} = &amp; {\\frac{{M}_{xz}}{m}} = {\\frac{81\/35}{108\/35}} = {\\frac{81}{108}} = {0.75}, \\\\\r\n{\\overline{z}} = &amp; {\\frac{{M}_{xy}}{m}} = {\\frac{54\/35}{108\/35}} = {\\frac{54}{108}} = {0.5}.\r\n\\end{aligned}[\/latex]<\/p>\r\nThe center of mass for the tetrahedron [latex]Q[\/latex] is the point [latex]{({2.25},{0.75},{0.5})}.[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConsider the same region [latex]Q[\/latex] (Figure 1) and use the density function [latex]{\\rho}{({x},{y},{z})} = xy^{2}z[\/latex]. Find the center of mass.\r\n\r\n[reveal-answer q=\"039842756\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"039842756\"]\r\n\r\n[latex]\\left(\\frac32,\\frac98,\\frac12\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to the above Try It[\/caption]\r\n\r\n<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8197113&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=w7ayXgDTBiM&amp;video_target=tpm-plugin-opzew3rd-w7ayXgDTBiM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP5.41_transcript.html\">transcript for \u201cCP 5.41\u201d here (opens in new window).<\/a><\/center>\r\n<p id=\"fs-id1167793277636\">We conclude this section with an example of finding moments of inertia [latex]I_x[\/latex],\u00a0[latex]I_y[\/latex], and\u00a0[latex]I_z[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1167793277667\" class=\"ui-has-child-title\" data-type=\"example\">\r\n<div class=\"textbox exercises\">\r\n<h3>example: finding the moments of inertia of a solid<\/h3>\r\nSuppose that [latex]Q[\/latex] is a solid region and is bounded by [latex]x+2y+3z=6[\/latex] and the coordinate planes with density [latex]{\\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[\/latex] (see\u00a0Figure 1). Find the moments of inertia of the tetrahedron [latex]Q[\/latex] about the [latex]yz[\/latex]-plane, the [latex]xz[\/latex]-plane, and the [latex]xy[\/latex]-plane.\r\n\r\n[reveal-answer q=\"452110456\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"452110456\"]\r\nOnce again, we can almost immediately write the limits of integration and hence we can quickly proceed to evaluating the moments of inertia. Using the formula stated before, the moments of inertia of the tetrahedron [latex]Q[\/latex] about the [latex]xy[\/latex]-plane, the [latex]xz[\/latex]-plane, and the [latex]yz[\/latex]-plane are\r\n\r\n<center>[latex]\\begin{aligned}\r\n{{I}_{x}} = &amp; {\\underset{Q}{\\displaystyle\\iiint}}{({{y}^{2}} + {{z}^{2}})}{\\rho}{({x},{y},{z})}{dV}, \\\\\r\n{{I}_{y}} = &amp; {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{z}^{2}})}{\\rho}{({x},{y},{z})}{dV},\r\n\\end{aligned}[\/latex]<\/center>\r\nand\r\n\r\n<center>[latex]{{I}_{z}} = {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{y}^{2}})}{\\rho}{({x},{y},{z})}{dV} \\ {\\text{with}} \\ {\\rho}{({x},{y},{z})} = {{x}^{2}}{yz}.[\/latex]<\/center>\r\nProceeding with the computations, we have\r\n\r\n<center>[latex]\\begin{align}<\/center>{{I}_{x}} &amp;= {\\underset{Q}{\\displaystyle\\iiint}}{({{y}^{2}} + {{z}^{2}})}x^2yz \\ {dV}=\\displaystyle\\int_{x=0}^{x=6}\\displaystyle\\int_{y=0}^{y=\\frac{1}{2}(6-x)}\\displaystyle\\int_{z=0}^{z=\\frac13(6-x-2y)}(y^2+z^2)x^2yz \\ dz \\ dy \\ dx = \\frac{117}{35}\\approx3.343, \\\\\r\n\r\n{{I}_{y}} &amp;= {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{z}^{2}})}x^2yz \\ {dV}=\\displaystyle\\int_{x=0}^{x=6}\\displaystyle\\int_{y=0}^{y=\\frac{1}{2}(6-x)}\\displaystyle\\int_{z=0}^{z=\\frac13(6-x-2y)}(x^2+z^2)x^2yz \\ dz \\ dy \\ dx = \\frac{684}{35}\\approx19.543, \\\\\r\n\r\n{{I}_{z}} &amp;= {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{y}^{2}})}x^2yz \\ {dV}=\\displaystyle\\int_{x=0}^{x=6}\\displaystyle\\int_{y=0}^{y=\\frac{1}{2}(6-x)}\\displaystyle\\int_{z=0}^{z=\\frac13(6-x-2y)}(x^2+y^2)x^2yz \\ dz \\ dy \\ dx = \\frac{729}{35}\\approx20.829.\r\n\r\n\\end{align}\r\n\r\n[\/latex]\r\n\r\nThus, the moments of inertia of the tetrahedron [latex]Q[\/latex] about the\u00a0[latex]yz[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the\u00a0[latex]xy[\/latex]-plane are [latex]{117\/35}, {684\/35},[\/latex] and [latex]{729\/35}[\/latex], respectively.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nConsider the same region [latex]Q[\/latex] (Figure 1), and use the density function [latex]{\\rho}{({x},{y},{z})} = {x}{{y}^{2}}{z}[\/latex]. Find the moments of inertia about the three coordinate planes.\r\n\r\n[reveal-answer q=\"485732209\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"485732209\"]\r\n\r\nThe moments of inertia of the tetrahedron [latex]Q[\/latex] about the [latex]yz[\/latex]-plane, the [latex]xz[\/latex]-plane, and the [latex]xy[\/latex]-plane are<span style=\"white-space: nowrap;\"> [latex]99\/35[\/latex],\u00a0[latex]36\/7[\/latex], and\u00a0[latex]243\/35[\/latex],\u00a0<\/span>respectively.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to the above Try It[\/caption]\r\n\r\n<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8197112&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=_EopC8p7DmM&amp;video_target=tpm-plugin-vi6rhg0g-_EopC8p7DmM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP5.42_transcript.html\">transcript for \u201cCP 5.42\u201d here (opens in new window).<\/a><\/center>&nbsp;\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3 style=\"text-align: center;\">Learning Objectives<\/h3>\n<ul class=\"os-abstract\">\n<li><span class=\"os-abstract-content\">Use triple integrals to locate the center of mass of a three-dimensional object.<\/span><\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1167793294160\">All the expressions of double integrals discussed so far can be modified to become triple integrals.<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">definition<\/h3>\n<hr \/>\n<p id=\"fs-id1167793294168\">If we have a solid object [latex]Q[\/latex]\u00a0with a density function [latex]{\\rho}{({x},{y},{z})}[\/latex] at any point[latex](x, y, z)[\/latex] in space, then its mass is<\/p>\n<p style=\"text-align: center;\">[latex]{m} = {\\underset{Q}{\\displaystyle\\iiint}}{\\rho}{({x},{y},{z})}{dV}.[\/latex]<\/p>\n<p id=\"fs-id1167793252305\">Its moments about the [latex]xy[\/latex]-plane, the [latex]xz[\/latex]-plane, and the [latex]yz[\/latex]-plane are<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}  {{M}_{xy}} = & {\\underset{Q}{\\displaystyle\\iiint}}{z}{\\rho}{({x},{y},{z})}{dV}, {{M}_{xz}} = {\\underset{Q}{\\displaystyle\\iiint}}{y}{\\rho}{({x},{y},{z})}{dV}, \\\\  {{M}_{yz}} = & {\\underset{Q}{\\displaystyle\\iiint}}{x}{\\rho}{({x},{y},{z})}{dV}.  \\end{aligned}[\/latex]<\/p>\n<p id=\"fs-id1167793499101\">If the center of mass of the object is the point [latex]{\\left ( {\\overline{x}},{\\overline{y}},{\\overline{z}} \\right )},[\/latex] then<\/p>\n<p style=\"text-align: center;\">[latex]\\large{{\\overline{x}} = {\\dfrac{{M}_{yz}}{m}}, \\ {\\overline{y}} = {\\dfrac{{M}_{xz}}{m}}, \\ {\\overline{z}} = {\\dfrac{{M}_{xy}}{m}}}.[\/latex]<\/p>\n<p id=\"fs-id1167794181252\">Also, if the solid object is homogeneous (with constant density), then the center of mass becomes the centroid of the solid. Finally, the moments of inertia about the [latex]yz[\/latex]-plane, the [latex]xz[\/latex]-plane,\u00a0<span id=\"MathJax-Element-184-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; overflow: initial; display: inline-table; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;\/mi&gt;&lt;mi&gt;z&lt;\/mi&gt;&lt;mtext&gt;-plane,&lt;\/mtext&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;\/mi&gt;&lt;mi&gt;z&lt;\/mi&gt;&lt;mtext&gt;-plane,&lt;\/mtext&gt;&lt;\/mrow&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt;\"><\/span>and the [latex]xy[\/latex]-plane are<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}  {{I}_{x}} = & {\\underset{Q}{\\displaystyle\\iiint}}{({{y}^{2}} + {{z}^{2}})}{\\rho}{({x},{y},{z})}{dV}, \\\\  {{I}_{y}} = &{\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{z}^{2}})}{\\rho}{({x},{y},{z})}{dV}, \\\\  {{I}_{z}} = & {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{y}^{2}})}{\\rho}{({x},{y},{z})}{dV}.  \\end{aligned}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: finding the mass of a solid<\/h3>\n<p>Suppose that [latex]Q[\/latex] is a solid region bounded by [latex]x+2y+3z=6[\/latex] and the coordinate planes and has density [latex]{\\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[\/latex]. Find the total mass.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q875204756\">Show Solution<\/span><\/p>\n<div id=\"q875204756\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793589656\">The region [latex]Q[\/latex] is a tetrahedron (Figure 1) meeting the axes at the points [latex](6, 0, 0)[\/latex], [latex](0, 3, 0)[\/latex], and [latex](0, 0, 2)[\/latex]. To find the limits of integration, let [latex]z=0[\/latex] in the slanted plane [latex]{z} = {\\frac{1}{3}}{({6} - {x} - {{2}{y}})}[\/latex]. Then for [latex]x[\/latex] and [latex]y[\/latex] find the projection of [latex]Q[\/latex] onto the [latex]xy[\/latex]-plane, which is bounded by the axes and the line [latex]x+2y=6[\/latex]. Hence the mass is<\/p>\n<div style=\"text-align: center;\">\n[latex]{m} = {\\underset{Q}{\\displaystyle\\iiint}}{\\rho}{({x},{y},{z})}{dV} = \\displaystyle\\int^{{x} = {6}}_{{x} = {0}} \\ \\displaystyle\\int^{y = \\frac{1}{2}({6} - x)}_{y = 0} \\ \\displaystyle\\int^{z = \\frac{1}{3}(6 - x - 2y)}_{z = 0} \\ x^{2}yz \\ dz \\ dy \\ dx = \\frac{108}{35} \\ \\approx \\ 3.086.[\/latex]<\/div>\n<div id=\"attachment_1414\" style=\"width: 278px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1414\" class=\"size-full wp-image-1414\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/26174918\/5-6-7.jpeg\" alt=\"In x y z space, the solid Q is shown with corners (0, 0, 0), (0, 0, 2), (0, 3, 0), and (6, 0, 0). Alternatively, you could consider the solid as being bounded by the x y, x z, and y z planes and the plane x + 2y + 3z = 6, forming an irregular tetrahedron.\" width=\"268\" height=\"309\" \/><\/p>\n<p id=\"caption-attachment-1414\" class=\"wp-caption-text\">Figure 1. Finding the mass of a three-dimensional solid [latex]Q[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Consider the same region [latex]Q[\/latex] (Figure 1), and use the density function [latex]{\\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[\/latex].\u00a0<span style=\"font-size: 1rem; text-align: initial;\">Find the mass.<\/span><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q685720429\">Show Solution<\/span><\/p>\n<div id=\"q685720429\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{54}{35}=1.543[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: finding the center of Mass of a solid<\/h3>\n<p>Suppose [latex]Q[\/latex] is a solid region bounded by the plane [latex]x+2y+3z=6[\/latex] and the coordinate planes with density [latex]{\\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[\/latex] (see\u00a0Figure 1). Find the center of mass using decimal approximation. Use the mass found in\u00a0Example &#8220;Finding the Mass of a Solid&#8221;.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q723096348\">Show Solution<\/span><\/p>\n<div id=\"q723096348\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793777311\">We have used this tetrahedron before and know the limits of integration, so we can proceed to the computations right away. First, we need to find the moments about the\u00a0[latex]xy[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the\u00a0[latex]yz[\/latex]-plane:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}  {{M}_{xy}} = &{\\underset{Q}{\\displaystyle\\iiint}}{z}{\\rho}{({x},{y},{z})}{dV} = {\\displaystyle\\int^{{x} = {6}}_{{x} = {0}}} \\ {\\displaystyle\\int^{{y} = {1\/2}{({6} - {x})}}_{{y} = {0}}} \\ {\\displaystyle\\int^{{z} = {1\/3}{({6} - {x} - {2y})}}_{{z} = {0}}} \\ {{x}^{2}}{{yz}^{2}}{dz} \\ {dy} \\ {dx} = {\\frac{54}{35}} \\ {\\approx} \\ {1.543}, \\\\  {{M}_{xz}} = & {\\underset{Q}{\\displaystyle\\iiint}}{y}{\\rho}{({x},{y},{z})}{dV} = {\\displaystyle\\int^{{x} = {6}}_{{x} = {0}}} \\ {\\displaystyle\\int^{{y} = {1\/2}{({6} - {x})}}_{{y} = {0}}} \\ {\\displaystyle\\int^{{z} = {1\/3}{({6} - {x} - {2y})}}_{{z} = {0}}} \\ {{x}^{2}}{{y}^{2}}{z} \\ {dz} \\ {dy} \\ {dx} = {\\frac{81}{35}} \\ {\\approx} \\ {2.314}, \\\\  {{M}_{yz}} = & {\\underset{Q}{\\displaystyle\\iiint}}{x}{\\rho}{({x},{y},{z})}{dV} = {\\displaystyle\\int^{{x} = {6}}_{{x} = {0}}} \\ {\\displaystyle\\int^{{y} = {1\/2}{({6} - {x})}}_{{y} = {0}}} \\ {\\displaystyle\\int^{{z} = {1\/3}{({6} - {x} - {2y})}}_{{z} = {0}}} \\ {{x}^{3}}{yz} \\ {dz} \\ {dy} \\ {dx} = {\\frac{243}{35}} \\ {\\approx} \\ {6.943}.  \\end{aligned}[\/latex]<\/p>\n<p>Hence the center of mass is<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}  {\\overline{x}} = & {\\frac{{M}_{yz}}{m}}, \\ {\\overline{y}} = {\\frac{{M}_{xz}}{m}}, \\ {\\overline{z}} = {\\frac{{M}_{xy}}{m}}, \\\\  {\\overline{x}} = & {\\frac{{M}_{yz}}{m}} = {\\frac{243\/35}{108\/35}} = {\\frac{243}{108}} = {2.25}, \\\\  {\\overline{y}} = & {\\frac{{M}_{xz}}{m}} = {\\frac{81\/35}{108\/35}} = {\\frac{81}{108}} = {0.75}, \\\\  {\\overline{z}} = & {\\frac{{M}_{xy}}{m}} = {\\frac{54\/35}{108\/35}} = {\\frac{54}{108}} = {0.5}.  \\end{aligned}[\/latex]<\/p>\n<p>The center of mass for the tetrahedron [latex]Q[\/latex] is the point [latex]{({2.25},{0.75},{0.5})}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Consider the same region [latex]Q[\/latex] (Figure 1) and use the density function [latex]{\\rho}{({x},{y},{z})} = xy^{2}z[\/latex]. Find the center of mass.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q039842756\">Show Solution<\/span><\/p>\n<div id=\"q039842756\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(\\frac32,\\frac98,\\frac12\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8197113&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=w7ayXgDTBiM&amp;video_target=tpm-plugin-opzew3rd-w7ayXgDTBiM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP5.41_transcript.html\">transcript for \u201cCP 5.41\u201d here (opens in new window).<\/a><\/div>\n<p id=\"fs-id1167793277636\">We conclude this section with an example of finding moments of inertia [latex]I_x[\/latex],\u00a0[latex]I_y[\/latex], and\u00a0[latex]I_z[\/latex].<\/p>\n<div id=\"fs-id1167793277667\" class=\"ui-has-child-title\" data-type=\"example\">\n<div class=\"textbox exercises\">\n<h3>example: finding the moments of inertia of a solid<\/h3>\n<p>Suppose that [latex]Q[\/latex] is a solid region and is bounded by [latex]x+2y+3z=6[\/latex] and the coordinate planes with density [latex]{\\rho}{({x},{y},{z})} = {{x}^{2}}{y}{z}[\/latex] (see\u00a0Figure 1). Find the moments of inertia of the tetrahedron [latex]Q[\/latex] about the [latex]yz[\/latex]-plane, the [latex]xz[\/latex]-plane, and the [latex]xy[\/latex]-plane.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q452110456\">Show Solution<\/span><\/p>\n<div id=\"q452110456\" class=\"hidden-answer\" style=\"display: none\">\nOnce again, we can almost immediately write the limits of integration and hence we can quickly proceed to evaluating the moments of inertia. Using the formula stated before, the moments of inertia of the tetrahedron [latex]Q[\/latex] about the [latex]xy[\/latex]-plane, the [latex]xz[\/latex]-plane, and the [latex]yz[\/latex]-plane are<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{aligned}  {{I}_{x}} = & {\\underset{Q}{\\displaystyle\\iiint}}{({{y}^{2}} + {{z}^{2}})}{\\rho}{({x},{y},{z})}{dV}, \\\\  {{I}_{y}} = & {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{z}^{2}})}{\\rho}{({x},{y},{z})}{dV},  \\end{aligned}[\/latex]<\/div>\n<p>and<\/p>\n<div style=\"text-align: center;\">[latex]{{I}_{z}} = {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{y}^{2}})}{\\rho}{({x},{y},{z})}{dV} \\ {\\text{with}} \\ {\\rho}{({x},{y},{z})} = {{x}^{2}}{yz}.[\/latex]<\/div>\n<p>Proceeding with the computations, we have<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}<\/div>\n<p>{{I}_{x}} &= {\\underset{Q}{\\displaystyle\\iiint}}{({{y}^{2}} + {{z}^{2}})}x^2yz \\ {dV}=\\displaystyle\\int_{x=0}^{x=6}\\displaystyle\\int_{y=0}^{y=\\frac{1}{2}(6-x)}\\displaystyle\\int_{z=0}^{z=\\frac13(6-x-2y)}(y^2+z^2)x^2yz \\ dz \\ dy \\ dx = \\frac{117}{35}\\approx3.343, \\\\    {{I}_{y}} &= {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{z}^{2}})}x^2yz \\ {dV}=\\displaystyle\\int_{x=0}^{x=6}\\displaystyle\\int_{y=0}^{y=\\frac{1}{2}(6-x)}\\displaystyle\\int_{z=0}^{z=\\frac13(6-x-2y)}(x^2+z^2)x^2yz \\ dz \\ dy \\ dx = \\frac{684}{35}\\approx19.543, \\\\    {{I}_{z}} &= {\\underset{Q}{\\displaystyle\\iiint}}{({{x}^{2}} + {{y}^{2}})}x^2yz \\ {dV}=\\displaystyle\\int_{x=0}^{x=6}\\displaystyle\\int_{y=0}^{y=\\frac{1}{2}(6-x)}\\displaystyle\\int_{z=0}^{z=\\frac13(6-x-2y)}(x^2+y^2)x^2yz \\ dz \\ dy \\ dx = \\frac{729}{35}\\approx20.829.    \\end{align}[\/latex]<\/p>\n<p>Thus, the moments of inertia of the tetrahedron [latex]Q[\/latex] about the\u00a0[latex]yz[\/latex]-plane, the\u00a0[latex]xz[\/latex]-plane, and the\u00a0[latex]xy[\/latex]-plane are [latex]{117\/35}, {684\/35},[\/latex] and [latex]{729\/35}[\/latex], respectively.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Consider the same region [latex]Q[\/latex] (Figure 1), and use the density function [latex]{\\rho}{({x},{y},{z})} = {x}{{y}^{2}}{z}[\/latex]. Find the moments of inertia about the three coordinate planes.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q485732209\">Show Solution<\/span><\/p>\n<div id=\"q485732209\" class=\"hidden-answer\" style=\"display: none\">\n<p>The moments of inertia of the tetrahedron [latex]Q[\/latex] about the [latex]yz[\/latex]-plane, the [latex]xz[\/latex]-plane, and the [latex]xy[\/latex]-plane are<span style=\"white-space: nowrap;\"> [latex]99\/35[\/latex],\u00a0[latex]36\/7[\/latex], and\u00a0[latex]243\/35[\/latex],\u00a0<\/span>respectively.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8197112&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=_EopC8p7DmM&amp;video_target=tpm-plugin-vi6rhg0g-_EopC8p7DmM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP5.42_transcript.html\">transcript for \u201cCP 5.42\u201d here (opens in new window).<\/a><\/div>\n<p>&nbsp;<\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1072\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>CP 5.41. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>CP 5.42. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":428269,"menu_order":26,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"CP 5.41\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"CP 5.42\",\"author\":\"Ryan 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