{"id":1207,"date":"2021-06-30T17:02:10","date_gmt":"2021-06-30T17:02:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/problem-set-moments-and-centers-of-mass\/"},"modified":"2021-11-17T02:19:31","modified_gmt":"2021-11-17T02:19:31","slug":"problem-set-moments-and-centers-of-mass","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/problem-set-moments-and-centers-of-mass\/","title":{"raw":"Problem Set: Moments and Centers of Mass","rendered":"Problem Set: Moments and Centers of Mass"},"content":{"raw":"

For the following exercises (1-6), calculate the center of mass for the collection of masses given.<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex]{m}_{1}=2[\/latex] at [latex]{x}_{1}=1[\/latex] and [latex]{m}_{2}=4[\/latex] at [latex]{x}_{2}=2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.\u00a0<\/strong>[latex]{m}_{1}=1[\/latex] at [latex]{x}_{1}=-1[\/latex] and [latex]{m}_{2}=3[\/latex] at [latex]{x}_{2}=2[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793553765\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793553765\"]\r\n

[latex]\\frac{5}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>[latex]m=3[\/latex] at [latex]x=0,1,2,6[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>Unit masses at [latex](x,y)=(1,0),(0,1),(1,1)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793420570\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793420570\"]\r\n

[latex](\\frac{2}{3},\\frac{2}{3})[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>[latex]{m}_{1}=1[\/latex] at [latex](1,0)[\/latex] and [latex]{m}_{2}=4[\/latex] at [latex](0,1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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6.\u00a0<\/strong>[latex]{m}_{1}=1[\/latex] at [latex](1,0)[\/latex] and [latex]{m}_{2}=3[\/latex] at [latex](2,2)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793499078\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793499078\"]\r\n

[latex](\\frac{7}{4},\\frac{3}{2})[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (7-16), compute the center of mass [latex]\\overline{x}.[\/latex]<\/p>\r\n\r\n

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7.\u00a0<\/strong>[latex]\\rho =1[\/latex] for [latex]x\\in (-1,3)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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8.\u00a0<\/strong>[latex]\\rho ={x}^{2}[\/latex] for [latex]x\\in (0,L)[\/latex]<\/p>\r\n[reveal-answer q=\"29983544\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"29983544\"]\r\n\r\n[latex]\\frac{3L}{4}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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9.\u00a0<\/strong>[latex]\\rho =1[\/latex] for [latex]x\\in (0,1)[\/latex] and [latex]\\rho =2[\/latex] for [latex]x\\in (1,2)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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10.\u00a0<\/strong>[latex]\\rho = \\sin x[\/latex] for [latex]x\\in (0,\\pi )[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167794146640\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794146640\"]\r\n

[latex]\\frac{\\pi }{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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11.\u00a0<\/strong>[latex]\\rho = \\cos x[\/latex] for [latex]x\\in (0,\\frac{\\pi }{2})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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12.\u00a0<\/strong>[latex]\\rho ={e}^{x}[\/latex] for [latex]x\\in (0,2)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793571577\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793571577\"]\r\n

[latex]\\frac{{e}^{2}+1}{{e}^{2}-1}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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13.\u00a0<\/strong>[latex]\\rho ={x}^{3}+x{e}^{\\text{\u2212}x}[\/latex] for [latex]x\\in (0,1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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14.\u00a0<\/strong>[latex]\\rho =x \\sin x[\/latex] for [latex]x\\in (0,\\pi )[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167794127157\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794127157\"]\r\n

[latex]\\frac{{\\pi }^{2}-4}{\\pi }[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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15.\u00a0<\/strong>[latex]\\rho =\\sqrt{x}[\/latex] for [latex]x\\in (1,4)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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16.\u00a0<\/strong>[latex]\\rho =\\text{ln}x[\/latex] for [latex]x\\in (1,e)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793616428\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793616428\"]\r\n

[latex]\\frac{1}{4}(1+{e}^{2})[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (17-), compute the center of mass [latex](\\overline{x},\\overline{y}).[\/latex] Use symmetry to help locate the center of mass whenever possible.<\/p>\r\n\r\n

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17.\u00a0<\/strong>[latex]\\rho =7[\/latex] in the square [latex]0\\le x\\le 1,[\/latex] [latex]0\\le y\\le 1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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18.\u00a0<\/strong>[latex]\\rho =3[\/latex] in the triangle with vertices [latex](0,0),[\/latex] [latex](a,0),[\/latex] and [latex](0,b)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793420964\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793420964\"]\r\n

[latex](\\frac{a}{3},\\frac{b}{3})[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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19.\u00a0<\/strong>[latex]\\rho =2[\/latex] for the region bounded by [latex]y= \\cos (x),[\/latex] [latex]y=\\text{\u2212} \\cos (x),[\/latex] [latex]x=-\\frac{\\pi }{2},[\/latex] and [latex]x=\\frac{\\pi }{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use a calculator to draw the region, then compute the center of mass [latex](\\overline{x},\\overline{y}).[\/latex] Use symmetry to help locate the center of mass whenever possible.<\/p>\r\n\r\n

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20. [T]<\/strong> The region bounded by [latex]y= \\cos (2x),[\/latex] [latex]x=-\\frac{\\pi }{4},[\/latex] and [latex]x=\\frac{\\pi }{4}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793978386\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793978386\"]\r\n

[latex](0,\\frac{\\pi }{8})[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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21. [T]<\/strong> The region between [latex]y=2{x}^{2},[\/latex] [latex]y=0,[\/latex] [latex]x=0,[\/latex] and [latex]x=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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22. [T]<\/strong> The region between [latex]y=\\frac{5}{4}{x}^{2}[\/latex] and [latex]y=5[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793504224\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793504224\"]\r\n

[latex](0,3)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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23. [T]<\/strong> Region between [latex]y=\\sqrt{x},[\/latex] [latex]y=\\text{ln}(x),[\/latex] [latex]x=1,[\/latex] and [latex]x=4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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24. [T]<\/strong> The region bounded by [latex]y=0,[\/latex] [latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167794054214\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794054214\"]\r\n

[latex](0,\\frac{4}{\\pi })[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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25. [T]<\/strong> The region bounded by [latex]y=0,[\/latex] [latex]x=0,[\/latex] and [latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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26. [T]<\/strong> The region bounded by [latex]y={x}^{2}[\/latex] and [latex]y={x}^{4}[\/latex] in the first quadrant<\/p>\r\n[reveal-answer q=\"fs-id1167793445686\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793445686\"]\r\n

[latex](\\frac{5}{8},\\frac{1}{3})[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use the theorem of Pappus to determine the volume of the shape.<\/p>\r\n\r\n

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27.\u00a0<\/strong>Rotating [latex]y=mx[\/latex] around the [latex]x[\/latex]-axis between [latex]x=0[\/latex] and [latex]x=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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28.\u00a0<\/strong>Rotating [latex]y=mx[\/latex] around the [latex]y[\/latex]-axis between [latex]x=0[\/latex] and [latex]x=1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793384845\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793384845\"]\r\n

[latex]\\frac{m\\pi }{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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29.\u00a0<\/strong>A general cone created by rotating a triangle with vertices [latex](0,0),[\/latex] [latex](a,0),[\/latex] and [latex](0,b)[\/latex] around the [latex]y[\/latex]-axis. Does your answer agree with the volume of a cone?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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30.\u00a0<\/strong>A general cylinder created by rotating a rectangle with vertices [latex](0,0),[\/latex] [latex](a,0),(0,b),[\/latex] and [latex](a,b)[\/latex] around the [latex]y[\/latex]-axis. Does your answer agree with the volume of a cylinder?<\/p>\r\n[reveal-answer q=\"fs-id1167793729501\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793729501\"][latex]\\pi {a}^{2}b[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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31.\u00a0<\/strong>A sphere created by rotating a semicircle with radius [latex]a[\/latex] around the [latex]y[\/latex]-axis. Does your answer agree with the volume of a sphere?<\/p>\r\n\r\n<\/div>\r\n

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area [latex]M[\/latex] and the centroid [latex](\\overline{x},\\overline{y})[\/latex] for the given shapes. Use symmetry to help locate the center of mass whenever possible.<\/p>\r\n\r\n

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32. [T]<\/strong> Quarter-circle: [latex]y=\\sqrt{1-{x}^{2}},[\/latex] [latex]y=0,[\/latex] and [latex]x=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793940505\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793940505\"]\r\n

[latex](\\frac{4}{3\\pi },\\frac{4}{3\\pi })[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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33. [T]<\/strong> Triangle: [latex]y=x,[\/latex] [latex]y=2-x,[\/latex] and [latex]y=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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34. [T]<\/strong> Lens: [latex]y={x}^{2}[\/latex] and [latex]y=x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167794222414\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794222414\"]\r\n

[latex](\\frac{1}{2},\\frac{2}{5})[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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35. [T]<\/strong> Ring: [latex]{y}^{2}+{x}^{2}=1[\/latex] and [latex]{y}^{2}+{x}^{2}=4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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36. [T]<\/strong> Half-ring: [latex]{y}^{2}+{x}^{2}=1,[\/latex] [latex]{y}^{2}+{x}^{2}=4,[\/latex] and [latex]y=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167794212219\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794212219\"]\r\n

[latex](0,\\frac{28}{9\\pi })[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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37.\u00a0<\/strong>Find the generalized center of mass in the sliver between [latex]y={x}^{a}[\/latex] and [latex]y={x}^{b}[\/latex] with [latex]a>b.[\/latex] Then, use the Pappus theorem to find the volume of the solid generated when revolving around the [latex]y[\/latex]-axis.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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38.\u00a0<\/strong>Find the generalized center of mass between [latex]y={a}^{2}-{x}^{2},[\/latex] [latex]x=0,[\/latex] and [latex]y=0.[\/latex] Then, use the Pappus theorem to find the volume of the solid generated when revolving around the [latex]y[\/latex]-axis.<\/p>\r\n[reveal-answer q=\"fs-id1167793521416\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793521416\"]\r\n

Center of mass: [latex](\\frac{a}{6},\\frac{4{a}^{2}}{5}),[\/latex] volume: [latex]\\frac{2\\pi {a}^{4}}{9}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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39.\u00a0<\/strong>Find the generalized center of mass between [latex]y=b \\sin (ax),[\/latex] [latex]x=0,[\/latex] and [latex]x=\\frac{\\pi }{a}.[\/latex] Then, use the Pappus theorem to find the volume of the solid generated when revolving around the [latex]y[\/latex]-axis.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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40.\u00a0<\/strong>Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius [latex]a[\/latex] is positioned with the left end of the circle at [latex]x=b,[\/latex] [latex]b>0,[\/latex] and is rotated around the [latex]y[\/latex]-axis.<\/p>\r\n\"This\r\n[reveal-answer q=\"fs-id1167793705297\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793705297\"]<\/span>\r\n\r\nVolume: [latex]2{\\pi }^{2}{a}^{2}(b+a)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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41.<\/strong> Find the center of mass [latex](\\overline{x},\\overline{y})[\/latex] for a thin wire along the semicircle [latex]y=\\sqrt{1-{x}^{2}}[\/latex] with unit mass.<\/p>\r\n(Hint: Use the theorem of Pappus.)<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

For the following exercises (1-6), calculate the center of mass for the collection of masses given.<\/p>\n

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1.\u00a0<\/strong>[latex]{m}_{1}=2[\/latex] at [latex]{x}_{1}=1[\/latex] and [latex]{m}_{2}=4[\/latex] at [latex]{x}_{2}=2[\/latex]<\/p>\n<\/div>\n<\/div>\n

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2.\u00a0<\/strong>[latex]{m}_{1}=1[\/latex] at [latex]{x}_{1}=-1[\/latex] and [latex]{m}_{2}=3[\/latex] at [latex]{x}_{2}=2[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\frac{5}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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3.\u00a0<\/strong>[latex]m=3[\/latex] at [latex]x=0,1,2,6[\/latex]<\/p>\n<\/div>\n<\/div>\n

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4.\u00a0<\/strong>Unit masses at [latex](x,y)=(1,0),(0,1),(1,1)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex](\\frac{2}{3},\\frac{2}{3})[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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5.\u00a0<\/strong>[latex]{m}_{1}=1[\/latex] at [latex](1,0)[\/latex] and [latex]{m}_{2}=4[\/latex] at [latex](0,1)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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6.\u00a0<\/strong>[latex]{m}_{1}=1[\/latex] at [latex](1,0)[\/latex] and [latex]{m}_{2}=3[\/latex] at [latex](2,2)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex](\\frac{7}{4},\\frac{3}{2})[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (7-16), compute the center of mass [latex]\\overline{x}.[\/latex]<\/p>\n

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7.\u00a0<\/strong>[latex]\\rho =1[\/latex] for [latex]x\\in (-1,3)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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8.\u00a0<\/strong>[latex]\\rho ={x}^{2}[\/latex] for [latex]x\\in (0,L)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\frac{3L}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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9.\u00a0<\/strong>[latex]\\rho =1[\/latex] for [latex]x\\in (0,1)[\/latex] and [latex]\\rho =2[\/latex] for [latex]x\\in (1,2)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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10.\u00a0<\/strong>[latex]\\rho = \\sin x[\/latex] for [latex]x\\in (0,\\pi )[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\frac{\\pi }{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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11.\u00a0<\/strong>[latex]\\rho = \\cos x[\/latex] for [latex]x\\in (0,\\frac{\\pi }{2})[\/latex]<\/p>\n<\/div>\n<\/div>\n

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12.\u00a0<\/strong>[latex]\\rho ={e}^{x}[\/latex] for [latex]x\\in (0,2)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\frac{{e}^{2}+1}{{e}^{2}-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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13.\u00a0<\/strong>[latex]\\rho ={x}^{3}+x{e}^{\\text{\u2212}x}[\/latex] for [latex]x\\in (0,1)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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14.\u00a0<\/strong>[latex]\\rho =x \\sin x[\/latex] for [latex]x\\in (0,\\pi )[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\frac{{\\pi }^{2}-4}{\\pi }[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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15.\u00a0<\/strong>[latex]\\rho =\\sqrt{x}[\/latex] for [latex]x\\in (1,4)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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16.\u00a0<\/strong>[latex]\\rho =\\text{ln}x[\/latex] for [latex]x\\in (1,e)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\frac{1}{4}(1+{e}^{2})[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (17-), compute the center of mass [latex](\\overline{x},\\overline{y}).[\/latex] Use symmetry to help locate the center of mass whenever possible.<\/p>\n

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17.\u00a0<\/strong>[latex]\\rho =7[\/latex] in the square [latex]0\\le x\\le 1,[\/latex] [latex]0\\le y\\le 1[\/latex]<\/p>\n<\/div>\n<\/div>\n

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18.\u00a0<\/strong>[latex]\\rho =3[\/latex] in the triangle with vertices [latex](0,0),[\/latex] [latex](a,0),[\/latex] and [latex](0,b)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex](\\frac{a}{3},\\frac{b}{3})[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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19.\u00a0<\/strong>[latex]\\rho =2[\/latex] for the region bounded by [latex]y= \\cos (x),[\/latex] [latex]y=\\text{\u2212} \\cos (x),[\/latex] [latex]x=-\\frac{\\pi }{2},[\/latex] and [latex]x=\\frac{\\pi }{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, use a calculator to draw the region, then compute the center of mass [latex](\\overline{x},\\overline{y}).[\/latex] Use symmetry to help locate the center of mass whenever possible.<\/p>\n

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20. [T]<\/strong> The region bounded by [latex]y= \\cos (2x),[\/latex] [latex]x=-\\frac{\\pi }{4},[\/latex] and [latex]x=\\frac{\\pi }{4}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex](0,\\frac{\\pi }{8})[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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21. [T]<\/strong> The region between [latex]y=2{x}^{2},[\/latex] [latex]y=0,[\/latex] [latex]x=0,[\/latex] and [latex]x=1[\/latex]<\/p>\n<\/div>\n<\/div>\n

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22. [T]<\/strong> The region between [latex]y=\\frac{5}{4}{x}^{2}[\/latex] and [latex]y=5[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex](0,3)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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23. [T]<\/strong> Region between [latex]y=\\sqrt{x},[\/latex] [latex]y=\\text{ln}(x),[\/latex] [latex]x=1,[\/latex] and [latex]x=4[\/latex]<\/p>\n<\/div>\n<\/div>\n

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24. [T]<\/strong> The region bounded by [latex]y=0,[\/latex] [latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex](0,\\frac{4}{\\pi })[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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25. [T]<\/strong> The region bounded by [latex]y=0,[\/latex] [latex]x=0,[\/latex] and [latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n

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26. [T]<\/strong> The region bounded by [latex]y={x}^{2}[\/latex] and [latex]y={x}^{4}[\/latex] in the first quadrant<\/p>\n

Show Solution<\/span><\/p>\n
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[latex](\\frac{5}{8},\\frac{1}{3})[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, use the theorem of Pappus to determine the volume of the shape.<\/p>\n

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27.\u00a0<\/strong>Rotating [latex]y=mx[\/latex] around the [latex]x[\/latex]-axis between [latex]x=0[\/latex] and [latex]x=1[\/latex]<\/p>\n<\/div>\n<\/div>\n

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28.\u00a0<\/strong>Rotating [latex]y=mx[\/latex] around the [latex]y[\/latex]-axis between [latex]x=0[\/latex] and [latex]x=1[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\frac{m\\pi }{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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29.\u00a0<\/strong>A general cone created by rotating a triangle with vertices [latex](0,0),[\/latex] [latex](a,0),[\/latex] and [latex](0,b)[\/latex] around the [latex]y[\/latex]-axis. Does your answer agree with the volume of a cone?<\/p>\n<\/div>\n<\/div>\n

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30.\u00a0<\/strong>A general cylinder created by rotating a rectangle with vertices [latex](0,0),[\/latex] [latex](a,0),(0,b),[\/latex] and [latex](a,b)[\/latex] around the [latex]y[\/latex]-axis. Does your answer agree with the volume of a cylinder?<\/p>\n

Show Solution<\/span><\/p>\n
[latex]\\pi {a}^{2}b[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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31.\u00a0<\/strong>A sphere created by rotating a semicircle with radius [latex]a[\/latex] around the [latex]y[\/latex]-axis. Does your answer agree with the volume of a sphere?<\/p>\n<\/div>\n

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area [latex]M[\/latex] and the centroid [latex](\\overline{x},\\overline{y})[\/latex] for the given shapes. Use symmetry to help locate the center of mass whenever possible.<\/p>\n

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32. [T]<\/strong> Quarter-circle: [latex]y=\\sqrt{1-{x}^{2}},[\/latex] [latex]y=0,[\/latex] and [latex]x=0[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex](\\frac{4}{3\\pi },\\frac{4}{3\\pi })[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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33. [T]<\/strong> Triangle: [latex]y=x,[\/latex] [latex]y=2-x,[\/latex] and [latex]y=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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34. [T]<\/strong> Lens: [latex]y={x}^{2}[\/latex] and [latex]y=x[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex](\\frac{1}{2},\\frac{2}{5})[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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35. [T]<\/strong> Ring: [latex]{y}^{2}+{x}^{2}=1[\/latex] and [latex]{y}^{2}+{x}^{2}=4[\/latex]<\/p>\n<\/div>\n<\/div>\n

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36. [T]<\/strong> Half-ring: [latex]{y}^{2}+{x}^{2}=1,[\/latex] [latex]{y}^{2}+{x}^{2}=4,[\/latex] and [latex]y=0[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex](0,\\frac{28}{9\\pi })[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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37.\u00a0<\/strong>Find the generalized center of mass in the sliver between [latex]y={x}^{a}[\/latex] and [latex]y={x}^{b}[\/latex] with [latex]a>b.[\/latex] Then, use the Pappus theorem to find the volume of the solid generated when revolving around the [latex]y[\/latex]-axis.<\/p>\n<\/div>\n<\/div>\n

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38.\u00a0<\/strong>Find the generalized center of mass between [latex]y={a}^{2}-{x}^{2},[\/latex] [latex]x=0,[\/latex] and [latex]y=0.[\/latex] Then, use the Pappus theorem to find the volume of the solid generated when revolving around the [latex]y[\/latex]-axis.<\/p>\n

Show Solution<\/span><\/p>\n
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Center of mass: [latex](\\frac{a}{6},\\frac{4{a}^{2}}{5}),[\/latex] volume: [latex]\\frac{2\\pi {a}^{4}}{9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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39.\u00a0<\/strong>Find the generalized center of mass between [latex]y=b \\sin (ax),[\/latex] [latex]x=0,[\/latex] and [latex]x=\\frac{\\pi }{a}.[\/latex] Then, use the Pappus theorem to find the volume of the solid generated when revolving around the [latex]y[\/latex]-axis.<\/p>\n<\/div>\n<\/div>\n

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40.\u00a0<\/strong>Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius [latex]a[\/latex] is positioned with the left end of the circle at [latex]x=b,[\/latex] [latex]b>0,[\/latex] and is rotated around the [latex]y[\/latex]-axis.<\/p>\n

\"This<\/p>\n

Show Solution<\/span><\/p>\n
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Volume: [latex]2{\\pi }^{2}{a}^{2}(b+a)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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41.<\/strong> Find the center of mass [latex](\\overline{x},\\overline{y})[\/latex] for a thin wire along the semicircle [latex]y=\\sqrt{1-{x}^{2}}[\/latex] with unit mass.<\/p>\n

(Hint: Use the theorem of Pappus.)<\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t

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