{"id":476,"date":"2025-02-13T19:45:19","date_gmt":"2025-02-13T19:45:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/moments-and-centers-of-mass-fresh-take\/"},"modified":"2025-02-13T19:45:19","modified_gmt":"2025-02-13T19:45:19","slug":"moments-and-centers-of-mass-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/moments-and-centers-of-mass-fresh-take\/","title":{"raw":"Moments and Centers of Mass: Fresh Take","rendered":"Moments and Centers of Mass: Fresh Take"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Find the balance point (center of mass) of straight objects and flat surfaces<\/li>\n\t<li>Utilize a shape\u2019s symmetry to find the centroid, or geometric center, of flat objects<\/li>\n\t<li>Use Pappus\u2019s theorem to calculate the volume of an object<\/li>\n<\/ul>\n<\/section>\n<h2>Center of Mass and Moments<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Center of Mass:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">The balancing point of an object or system<\/li>\n\t<li class=\"whitespace-normal break-words\">Point where the total mass can be concentrated without changing the object's behavior<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Moments:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li style=\"list-style-type: none;\">\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Measure of the rotational force applied to an object<\/li>\n\t<li class=\"whitespace-normal break-words\">First moment with respect to origin: [latex]M = \\sum_{i=1}^n m_i x_i[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">One-Dimensional Systems:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Center of mass: [latex]\\overline{x} = \\frac{\\sum_{i=1}^n m_i x_i}{\\sum_{i=1}^n m_i}[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Two-Dimensional Systems:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Moments: [latex]M_x = \\sum_{i=1}^n m_i y_i[\/latex], [latex]M_y = \\sum_{i=1}^n m_i x_i[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Center of mass: [latex]\\overline{x} = \\frac{M_y}{m}[\/latex], [latex]\\overline{y} = \\frac{M_x}{m}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Identify the system (point masses or continuous object)<\/li>\n\t<li class=\"whitespace-normal break-words\">Choose an appropriate coordinate system<\/li>\n\t<li class=\"whitespace-normal break-words\">Calculate the total mass of the system<\/li>\n\t<li class=\"whitespace-normal break-words\">Compute moments with respect to axes<\/li>\n\t<li class=\"whitespace-normal break-words\">Use formulas to find the center of mass coordinates<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the center of mass for three point masses:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]m_1 = 2\\text{ kg}[\/latex] at [latex](-1, 3)[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]m_2 = 6\\text{ kg}[\/latex] at [latex](1, 1)[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]m_3 = 4\\text{ kg}[\/latex] at [latex](2, -2)[\/latex]<\/li>\n<\/ul>\n<p><br>\n[reveal-answer q=\"42771\"]Show Answer[\/reveal-answer]<br>\n[hidden-answer a=\"42771\"]<\/p>\n<p>Total mass:<\/p>\n<p style=\"text-align: center;\">[latex]m = 2 + 6 + 4 = 12\\text{ kg}[\/latex]<\/p>\n<p>Moments:<\/p>\n<p style=\"text-align: center;\">[latex]M_y = (-1 \\cdot 2) + (1 \\cdot 6) + (2 \\cdot 4) = 12[\/latex]<br>\n[latex]M_x = (3 \\cdot 2) + (1 \\cdot 6) + (-2 \\cdot 4) = 4[\/latex]<\/p>\n<p>Center of mass:<\/p>\n<p style=\"text-align: center;\">[latex]\\overline{x} = \\frac{M_y}{m} = \\frac{12}{12} = 1[\/latex]<br>\n[latex]\\overline{y} = \\frac{M_x}{m} = \\frac{4}{12} = \\frac{1}{3}[\/latex]<\/p>\n<p>Center of mass:<\/p>\n<p style=\"text-align: center;\">[latex](1, \\frac{1}{3})[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1167794036513\">Suppose four point masses are placed on a number line as follows:<\/p>\n<div id=\"fs-id1167793838320\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cccc}{m}_{1}=12\\text{kg,}\\text{ placed at }{x}_{1}=-4\\text{m}\\hfill &amp; &amp; &amp; {m}_{2}=12\\text{kg,}\\text{ placed at }{x}_{2}=4\\text{m}\\hfill \\\\ {m}_{3}=30\\text{kg,}\\text{ placed at }{x}_{3}=2\\text{m}\\hfill &amp; &amp; &amp; {m}_{4}=6\\text{kg,}\\text{ placed at }{x}_{4}=-6\\text{m}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167794036841\">Find the moment of the system with respect to the origin and find the center of mass of the system.<\/p>\n\n[reveal-answer q=\"fs-id1167793834465\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793834465\"]\n\n<p id=\"fs-id1167793834465\">[latex]M=24,\\overline{x}=\\frac{2}{5}\\text{m}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1167793268284\">Suppose three point masses are placed on a number line as follows (assume coordinates are given in meters):<\/p>\n<div id=\"fs-id1167794027904\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}{m}_{1}=5\\text{kg, placed at}(-2,-3),\\hfill \\\\ {m}_{2}=3\\text{kg, placed at}(2,3),\\hfill \\\\ {m}_{3}=2\\text{kg, placed at}(-3,-2).\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167793270540\">Find the center of mass of the system.<\/p>\n\n[reveal-answer q=\"fs-id1167794040411\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167794040411\"][latex](-1,-1)[\/latex] m\n\n<p>&nbsp;<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/-XKii-JZrqw?controls=0&amp;start=482&amp;end=583&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.6MomentsAndCentersOfMass482to583_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"2.6 Moments and Centers of Mass\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<h2>Center of Mass of Thin Plates<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Lamina:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Thin sheet with uniform density<\/li>\n\t<li class=\"whitespace-normal break-words\">Mass distributed continuously across a 2D region<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Centroid:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Geometric center of a 2D shape<\/li>\n\t<li class=\"whitespace-normal break-words\">Coincides with center of mass for uniform density<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Key Formulas:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Mass: [latex]m = \\rho \\int_a^b f(x) dx[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-axis Moment: [latex]M_x = \\rho \\int_a^b \\frac{[f(x)]^2}{2} dx[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-axis Moment: [latex]M_y = \\rho \\int_a^b x f(x) dx[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Center of Mass: [latex]\\overline{x} = \\frac{M_y}{m}, \\overline{y} = \\frac{M_x}{m}[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Symmetry Principle:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">If a region is symmetric about a line, its centroid lies on that line<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Identify the region [latex]R[\/latex] and its bounding function [latex]f(x)[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Set up integrals for mass and moments ([latex]M_x[\/latex] and [latex]M_y[\/latex])<\/li>\n\t<li class=\"whitespace-normal break-words\">Evaluate the integrals<\/li>\n\t<li class=\"whitespace-normal break-words\">Calculate center of mass coordinates using [latex]M_x[\/latex], [latex]M_y[\/latex], and [latex]m[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Simplify and interpret the results<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">\n<p>Let [latex]R[\/latex] be the region bounded above by the graph of the function [latex]f(x)={x}^{2}[\/latex] and below by the [latex]x[\/latex]-axis over the interval [latex]\\left[0,2\\right].[\/latex] Find the centroid of the region.<\/p>\n\n[reveal-answer q=\"fs-id1167793912452\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793912452\"]\n\n<p id=\"fs-id1167793912452\">The centroid of the region is [latex](\\frac{3}{2},\\frac{6}{5}).[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/-XKii-JZrqw?controls=0&amp;start=1006&amp;end=1143&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.6MomentsAndCentersOfMass1006to1143_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"2.6 Moments and Centers of Mass\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Let <em>R<\/em> be the region bounded above by the graph of the function [latex]f(x)=6-{x}^{2}[\/latex] and below by the graph of the function [latex]g(x)=3-2x.[\/latex] Find the centroid of the region.<\/p>\n<p>[reveal-answer q=\"fs-id1167793308050\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793308050\"]<\/p>\n<p id=\"fs-id1167793308050\">The centroid of the region is [latex](1,\\frac{13}{5}).[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/-XKii-JZrqw?controls=0&amp;start=1473&amp;end=1704&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.6MomentsAndCentersOfMass1473to1704_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"2.6 Moments and Centers of Mass\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find the centroid of the region bounded by [latex]y = \\sqrt{x}[\/latex] and the [latex]x[\/latex]-axis from [latex]x = 0[\/latex] to [latex]x = 4[\/latex].<\/p>\n<p><br>\n[reveal-answer q=\"480090\"]Show Answer[\/reveal-answer]<br>\n[hidden-answer a=\"480090\"]<\/p>\n<p>Mass:<\/p>\n<p style=\"text-align: center;\">[latex]m = \\int_0^4 \\sqrt{x} dx = \\frac{2}{3}x^{3\/2}|_0^4 = \\frac{16}{3}[\/latex]<\/p>\n<p>Moments:<\/p>\n<p style=\"text-align: center;\">[latex]M_x = \\int_0^4 \\frac{x}{2} dx = \\frac{1}{4}x^2|_0^4 = 4[\/latex]<br>\n[latex]M_y = \\int_0^4 x\\sqrt{x} dx = \\frac{2}{5}x^{5\/2}|_0^4 = \\frac{64}{5}[\/latex]<\/p>\n<p>Center of Mass:<\/p>\n<p style=\"text-align: center;\">[latex]\\overline{x} = \\frac{M_y}{m} = \\frac{64\/5}{16\/3} = \\frac{12}{5}[\/latex]<br>\n[latex]\\overline{y} = \\frac{M_x}{m} = \\frac{4}{16\/3} = \\frac{3}{4}[\/latex]<\/p>\n<p>Centroid:<\/p>\n<p style=\"text-align: center;\">[latex](\\frac{12}{5}, \\frac{3}{4})[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<h2>Center of Mass of a Region Bounded by Two Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Region Definition:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Bounded above by function [latex]f(x)[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Bounded below by function [latex]g(x)[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Bounded on sides by [latex]x = a[\/latex] and [latex]x = b[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Key Formulas:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Mass: [latex]m = \\rho \\int_a^b [f(x) - g(x)] dx[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-axis Moment: [latex]M_x = \\rho \\int_a^b \\frac{1}{2}([f(x)]^2 - [g(x)]^2) dx[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-axis Moment: [latex]M_y = \\rho \\int_a^b x[f(x) - g(x)] dx[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Center of Mass: [latex]\\overline{x} = \\frac{M_y}{m}, \\overline{y} = \\frac{M_x}{m}[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Centroid of Rectangle:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Located at [latex](x_i^<em>, \\frac{f(x_i^<\/em>) + g(x_i^*)}{2})[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Identify bounding functions [latex]f(x)[\/latex] and [latex]g(x)[\/latex] and integration limits<\/li>\n\t<li class=\"whitespace-normal break-words\">Set up integrals for mass and moments ([latex]M_x[\/latex] and [latex]M_y[\/latex])<\/li>\n\t<li class=\"whitespace-normal break-words\">Evaluate the integrals, often requiring integration by parts or substitution<\/li>\n\t<li class=\"whitespace-normal break-words\">Calculate center of mass coordinates using [latex]M_x[\/latex], [latex]M_y[\/latex], and [latex]m[\/latex]<\/li>\n\t<li class=\"whitespace-normal break-words\">Simplify and interpret the results<\/li>\n<\/ol>\n<\/div>\n<h2>The Symmetry Principle<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n\t<li class=\"whitespace-normal break-words\">Symmetry Principle:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">If a region is symmetric about a line, its centroid lies on that line<\/li>\n\t<li class=\"whitespace-normal break-words\">Simplifies calculations for symmetric shapes<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Theorem of Pappus for Volume:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Volume = Area of region \u00d7 Distance traveled by centroid<\/li>\n\t<li class=\"whitespace-normal break-words\">Applies to solids of revolution<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Identify symmetry in the given region<\/li>\n\t<li class=\"whitespace-normal break-words\">Use symmetry to simplify centroid calculations<\/li>\n\t<li class=\"whitespace-normal break-words\">For solids of revolution:\n\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Calculate the area of the region<\/li>\n\t<li class=\"whitespace-normal break-words\">Determine the path of the centroid<\/li>\n\t<li class=\"whitespace-normal break-words\">Apply the Theorem of Pappus<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1167793619908\">Let <em>R<\/em> be the region bounded above by the graph of the function [latex]f(x)=1-{x}^{2}[\/latex] and below by [latex]x[\/latex]-axis. Find the centroid of the region.<\/p>\n<p>[reveal-answer q=\"fs-id1167794075561\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167794075561\"]<\/p>\n<p id=\"fs-id1167794075561\">The centroid of the region is [latex](0,2\\text{\/}5).[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Let [latex]R[\/latex] be a circle of radius [latex]1[\/latex] centered at [latex](3,0).[\/latex] Use the theorem of Pappus for volume to find the volume of the torus generated by revolving [latex]R[\/latex] around the [latex]y[\/latex]-axis.<\/p>\n<p>[reveal-answer q=\"fs-id1167793504036\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793504036\"]<\/p>\n<p id=\"fs-id1167793504036\">[latex]6{\\pi }^{2}[\/latex] units<sup>3<\/sup><\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/-XKii-JZrqw?controls=0&amp;start=2076&amp;end=2127&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.6MomentsAndCentersOfMass2076to2127_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"2.6 Moments and Centers of Mass\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find the volume of a washer-shaped solid formed by revolving the region between concentric circles of radii [latex]3[\/latex] and [latex]5[\/latex] centered at [latex](4,0)[\/latex] around the [latex]y[\/latex]-axis.<\/p>\n<p><br>\n[reveal-answer q=\"766778\"]Show Answer[\/reveal-answer]<br>\n[hidden-answer a=\"766778\"]<\/p>\n<p>Area of the region:<\/p>\n<p style=\"text-align: center;\">[latex]A = \\pi(5^2 - 3^2) = 16\\pi[\/latex]<\/p>\n<p>Centroid location: Due to symmetry, centroid is at [latex](4,0)[\/latex]<\/p>\n<p>Distance traveled by centroid:<\/p>\n<p style=\"text-align: center;\">[latex]d = 2\\pi(4) = 8\\pi[\/latex]<\/p>\n<p>Volume using Pappus's Theorem:<\/p>\n<p style=\"text-align: center;\">[latex]V = A \\cdot d = 16\\pi \\cdot 8\\pi = 128\\pi^2[\/latex]<\/p>\n<p>The volume of the washer-shaped solid is [latex]128\\pi^2[\/latex] cubic units.<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Find the balance point (center of mass) of straight objects and flat surfaces<\/li>\n<li>Utilize a shape\u2019s symmetry to find the centroid, or geometric center, of flat objects<\/li>\n<li>Use Pappus\u2019s theorem to calculate the volume of an object<\/li>\n<\/ul>\n<\/section>\n<h2>Center of Mass and Moments<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Center of Mass:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The balancing point of an object or system<\/li>\n<li class=\"whitespace-normal break-words\">Point where the total mass can be concentrated without changing the object&#8217;s behavior<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Moments:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li style=\"list-style-type: none;\">\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Measure of the rotational force applied to an object<\/li>\n<li class=\"whitespace-normal break-words\">First moment with respect to origin: [latex]M = \\sum_{i=1}^n m_i x_i[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">One-Dimensional Systems:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Center of mass: [latex]\\overline{x} = \\frac{\\sum_{i=1}^n m_i x_i}{\\sum_{i=1}^n m_i}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Two-Dimensional Systems:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Moments: [latex]M_x = \\sum_{i=1}^n m_i y_i[\/latex], [latex]M_y = \\sum_{i=1}^n m_i x_i[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Center of mass: [latex]\\overline{x} = \\frac{M_y}{m}[\/latex], [latex]\\overline{y} = \\frac{M_x}{m}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the system (point masses or continuous object)<\/li>\n<li class=\"whitespace-normal break-words\">Choose an appropriate coordinate system<\/li>\n<li class=\"whitespace-normal break-words\">Calculate the total mass of the system<\/li>\n<li class=\"whitespace-normal break-words\">Compute moments with respect to axes<\/li>\n<li class=\"whitespace-normal break-words\">Use formulas to find the center of mass coordinates<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the center of mass for three point masses:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]m_1 = 2\\text{ kg}[\/latex] at [latex](-1, 3)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]m_2 = 6\\text{ kg}[\/latex] at [latex](1, 1)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]m_3 = 4\\text{ kg}[\/latex] at [latex](2, -2)[\/latex]<\/li>\n<\/ul>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q42771\">Show Answer<\/button><\/p>\n<div id=\"q42771\" class=\"hidden-answer\" style=\"display: none\">\n<p>Total mass:<\/p>\n<p style=\"text-align: center;\">[latex]m = 2 + 6 + 4 = 12\\text{ kg}[\/latex]<\/p>\n<p>Moments:<\/p>\n<p style=\"text-align: center;\">[latex]M_y = (-1 \\cdot 2) + (1 \\cdot 6) + (2 \\cdot 4) = 12[\/latex]<br \/>\n[latex]M_x = (3 \\cdot 2) + (1 \\cdot 6) + (-2 \\cdot 4) = 4[\/latex]<\/p>\n<p>Center of mass:<\/p>\n<p style=\"text-align: center;\">[latex]\\overline{x} = \\frac{M_y}{m} = \\frac{12}{12} = 1[\/latex]<br \/>\n[latex]\\overline{y} = \\frac{M_x}{m} = \\frac{4}{12} = \\frac{1}{3}[\/latex]<\/p>\n<p>Center of mass:<\/p>\n<p style=\"text-align: center;\">[latex](1, \\frac{1}{3})[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1167794036513\">Suppose four point masses are placed on a number line as follows:<\/p>\n<div id=\"fs-id1167793838320\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{cccc}{m}_{1}=12\\text{kg,}\\text{ placed at }{x}_{1}=-4\\text{m}\\hfill & & & {m}_{2}=12\\text{kg,}\\text{ placed at }{x}_{2}=4\\text{m}\\hfill \\\\ {m}_{3}=30\\text{kg,}\\text{ placed at }{x}_{3}=2\\text{m}\\hfill & & & {m}_{4}=6\\text{kg,}\\text{ placed at }{x}_{4}=-6\\text{m}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167794036841\">Find the moment of the system with respect to the origin and find the center of mass of the system.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793834465\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793834465\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793834465\">[latex]M=24,\\overline{x}=\\frac{2}{5}\\text{m}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1167793268284\">Suppose three point masses are placed on a number line as follows (assume coordinates are given in meters):<\/p>\n<div id=\"fs-id1167794027904\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{c}{m}_{1}=5\\text{kg, placed at}(-2,-3),\\hfill \\\\ {m}_{2}=3\\text{kg, placed at}(2,3),\\hfill \\\\ {m}_{3}=2\\text{kg, placed at}(-3,-2).\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1167793270540\">Find the center of mass of the system.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167794040411\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167794040411\" class=\"hidden-answer\" style=\"display: none\">[latex](-1,-1)[\/latex] m<\/p>\n<p>&nbsp;<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/-XKii-JZrqw?controls=0&amp;start=482&amp;end=583&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.6MomentsAndCentersOfMass482to583_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;2.6 Moments and Centers of Mass&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<h2>Center of Mass of Thin Plates<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Lamina:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Thin sheet with uniform density<\/li>\n<li class=\"whitespace-normal break-words\">Mass distributed continuously across a 2D region<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Centroid:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Geometric center of a 2D shape<\/li>\n<li class=\"whitespace-normal break-words\">Coincides with center of mass for uniform density<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Formulas:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Mass: [latex]m = \\rho \\int_a^b f(x) dx[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-axis Moment: [latex]M_x = \\rho \\int_a^b \\frac{[f(x)]^2}{2} dx[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-axis Moment: [latex]M_y = \\rho \\int_a^b x f(x) dx[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Center of Mass: [latex]\\overline{x} = \\frac{M_y}{m}, \\overline{y} = \\frac{M_x}{m}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Symmetry Principle:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If a region is symmetric about a line, its centroid lies on that line<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the region [latex]R[\/latex] and its bounding function [latex]f(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Set up integrals for mass and moments ([latex]M_x[\/latex] and [latex]M_y[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Evaluate the integrals<\/li>\n<li class=\"whitespace-normal break-words\">Calculate center of mass coordinates using [latex]M_x[\/latex], [latex]M_y[\/latex], and [latex]m[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Simplify and interpret the results<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">\n<p>Let [latex]R[\/latex] be the region bounded above by the graph of the function [latex]f(x)={x}^{2}[\/latex] and below by the [latex]x[\/latex]-axis over the interval [latex]\\left[0,2\\right].[\/latex] Find the centroid of the region.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793912452\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793912452\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793912452\">The centroid of the region is [latex](\\frac{3}{2},\\frac{6}{5}).[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/-XKii-JZrqw?controls=0&amp;start=1006&amp;end=1143&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.6MomentsAndCentersOfMass1006to1143_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;2.6 Moments and Centers of Mass&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Let <em>R<\/em> be the region bounded above by the graph of the function [latex]f(x)=6-{x}^{2}[\/latex] and below by the graph of the function [latex]g(x)=3-2x.[\/latex] Find the centroid of the region.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793308050\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793308050\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793308050\">The centroid of the region is [latex](1,\\frac{13}{5}).[\/latex]<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/-XKii-JZrqw?controls=0&amp;start=1473&amp;end=1704&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.6MomentsAndCentersOfMass1473to1704_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;2.6 Moments and Centers of Mass&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find the centroid of the region bounded by [latex]y = \\sqrt{x}[\/latex] and the [latex]x[\/latex]-axis from [latex]x = 0[\/latex] to [latex]x = 4[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q480090\">Show Answer<\/button><\/p>\n<div id=\"q480090\" class=\"hidden-answer\" style=\"display: none\">\n<p>Mass:<\/p>\n<p style=\"text-align: center;\">[latex]m = \\int_0^4 \\sqrt{x} dx = \\frac{2}{3}x^{3\/2}|_0^4 = \\frac{16}{3}[\/latex]<\/p>\n<p>Moments:<\/p>\n<p style=\"text-align: center;\">[latex]M_x = \\int_0^4 \\frac{x}{2} dx = \\frac{1}{4}x^2|_0^4 = 4[\/latex]<br \/>\n[latex]M_y = \\int_0^4 x\\sqrt{x} dx = \\frac{2}{5}x^{5\/2}|_0^4 = \\frac{64}{5}[\/latex]<\/p>\n<p>Center of Mass:<\/p>\n<p style=\"text-align: center;\">[latex]\\overline{x} = \\frac{M_y}{m} = \\frac{64\/5}{16\/3} = \\frac{12}{5}[\/latex]<br \/>\n[latex]\\overline{y} = \\frac{M_x}{m} = \\frac{4}{16\/3} = \\frac{3}{4}[\/latex]<\/p>\n<p>Centroid:<\/p>\n<p style=\"text-align: center;\">[latex](\\frac{12}{5}, \\frac{3}{4})[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Center of Mass of a Region Bounded by Two Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Region Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Bounded above by function [latex]f(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Bounded below by function [latex]g(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Bounded on sides by [latex]x = a[\/latex] and [latex]x = b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Formulas:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Mass: [latex]m = \\rho \\int_a^b [f(x) - g(x)] dx[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x[\/latex]-axis Moment: [latex]M_x = \\rho \\int_a^b \\frac{1}{2}([f(x)]^2 - [g(x)]^2) dx[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y[\/latex]-axis Moment: [latex]M_y = \\rho \\int_a^b x[f(x) - g(x)] dx[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Center of Mass: [latex]\\overline{x} = \\frac{M_y}{m}, \\overline{y} = \\frac{M_x}{m}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Centroid of Rectangle:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Located at [latex](x_i^<em>, \\frac{f(x_i^<\/em>) + g(x_i^*)}{2})[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify bounding functions [latex]f(x)[\/latex] and [latex]g(x)[\/latex] and integration limits<\/li>\n<li class=\"whitespace-normal break-words\">Set up integrals for mass and moments ([latex]M_x[\/latex] and [latex]M_y[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Evaluate the integrals, often requiring integration by parts or substitution<\/li>\n<li class=\"whitespace-normal break-words\">Calculate center of mass coordinates using [latex]M_x[\/latex], [latex]M_y[\/latex], and [latex]m[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Simplify and interpret the results<\/li>\n<\/ol>\n<\/div>\n<h2>The Symmetry Principle<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Symmetry Principle:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If a region is symmetric about a line, its centroid lies on that line<\/li>\n<li class=\"whitespace-normal break-words\">Simplifies calculations for symmetric shapes<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Theorem of Pappus for Volume:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Volume = Area of region \u00d7 Distance traveled by centroid<\/li>\n<li class=\"whitespace-normal break-words\">Applies to solids of revolution<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify symmetry in the given region<\/li>\n<li class=\"whitespace-normal break-words\">Use symmetry to simplify centroid calculations<\/li>\n<li class=\"whitespace-normal break-words\">For solids of revolution:\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Calculate the area of the region<\/li>\n<li class=\"whitespace-normal break-words\">Determine the path of the centroid<\/li>\n<li class=\"whitespace-normal break-words\">Apply the Theorem of Pappus<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1167793619908\">Let <em>R<\/em> be the region bounded above by the graph of the function [latex]f(x)=1-{x}^{2}[\/latex] and below by [latex]x[\/latex]-axis. Find the centroid of the region.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167794075561\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167794075561\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794075561\">The centroid of the region is [latex](0,2\\text{\/}5).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Let [latex]R[\/latex] be a circle of radius [latex]1[\/latex] centered at [latex](3,0).[\/latex] Use the theorem of Pappus for volume to find the volume of the torus generated by revolving [latex]R[\/latex] around the [latex]y[\/latex]-axis.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793504036\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793504036\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793504036\">[latex]6{\\pi }^{2}[\/latex] units<sup>3<\/sup><\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/-XKii-JZrqw?controls=0&amp;start=2076&amp;end=2127&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.6MomentsAndCentersOfMass2076to2127_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;2.6 Moments and Centers of Mass&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find the volume of a washer-shaped solid formed by revolving the region between concentric circles of radii [latex]3[\/latex] and [latex]5[\/latex] centered at [latex](4,0)[\/latex] around the [latex]y[\/latex]-axis.<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q766778\">Show Answer<\/button><\/p>\n<div id=\"q766778\" class=\"hidden-answer\" style=\"display: none\">\n<p>Area of the region:<\/p>\n<p style=\"text-align: center;\">[latex]A = \\pi(5^2 - 3^2) = 16\\pi[\/latex]<\/p>\n<p>Centroid location: Due to symmetry, centroid is at [latex](4,0)[\/latex]<\/p>\n<p>Distance traveled by centroid:<\/p>\n<p style=\"text-align: center;\">[latex]d = 2\\pi(4) = 8\\pi[\/latex]<\/p>\n<p>Volume using Pappus&#8217;s Theorem:<\/p>\n<p style=\"text-align: center;\">[latex]V = A \\cdot d = 16\\pi \\cdot 8\\pi = 128\\pi^2[\/latex]<\/p>\n<p>The volume of the washer-shaped solid is [latex]128\\pi^2[\/latex] cubic units.<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":17,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":450,"module-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\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/476"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/476\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/450"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/476\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=476"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=476"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=476"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=476"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}