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6.6E: Exercises for Section 6.6

  • Page ID
    70167
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    In exercises 1 - 6, calculate the center of mass for the collection of masses given.

    1) \(m_1=2\) at \(x_1=1\) and \(m_2=4\) at \(x_2=2\)

    2) \(m_1=1\) at \(x_1=−1\) and \(m_2=3\) at \(x_2=2\)

    Answer
    \(x = \frac{5}{4}\)

    3) \(m=3\) at \(x=0,1,2,6\)

    4) Unit masses at \((x,y)=(1,0),(0,1),(1,1)\)

    Answer
    \(\left(\frac{2}{3},\, \frac{2}{3}\right)\)

    5) \(m_1=1\) at \((1,0)\) and \(m_2=4\) at \((0,1)\)

    6) \(m_1=1\) at \((1,0)\) and \(m_2=3\) at \((2,2)\)

    Answer
    \(\left(\frac{7}{4},\,\frac{3}{2}\right)\)

    In exercises 7 - 16, compute the center of mass \(\bar x.\)

    7) \(ρ=1\) for \(x∈(−1,3)\)

    8) \(ρ=x^2\) for \(x∈(0,L)\)

    Answer
    \(\dfrac{3L}{4}\)

    9) \(ρ=1\) for \(x∈(0,1)\) and \(ρ=2\) for \(x∈(1,2)\)

    10) \(ρ=\sin x\) for \(x∈(0,π)\)

    Answer
    \(\frac{π}{2}\)

    11) \(ρ=\cos x\) for \(x∈\left(0,\frac{π}{2}\right)\)

    12) \(ρ=e^x\) for \(x∈(0,2)\)

    Answer
    \(\dfrac{e^2+1}{e^2−1}\)

    13) \(ρ=x^3+xe^{−x}\) for \(x∈(0,1)\)

    14) \(ρ=x\sin x\) for \(x∈(0,π)\)

    Answer
    \(\dfrac{π^2−4}{π}\)

    15) \(ρ=\sqrt{x}\) for \(x∈(1,4)\)

    16) \(ρ=\ln x\) for \(x∈(1,e)\)

    Answer
    \(\frac{1}{4}(1+e^2)\)

    In exercises 17 - 19, compute the center of mass \((\bar{x},\bar{y}).\) Use symmetry to help locate the center of mass whenever possible.

    17) \(ρ=7\) in the square \(0≤x≤1, \; 0≤y≤1\)

    18) \(ρ=3\) in the triangle with vertices \((0,0), \, (a,0)\), and \((0,b)\)

    Answer
    \(\left(\frac{a}{3},\, \frac{b}{3}\right)\)

    19) \(ρ=2\) for the region bounded by \(y=\cos(x), \; y=−\cos(x), \; x=−\frac{π}{2}\), and \(x=\frac{π}{2}\)

    In exercises 20 - 26, use a calculator to draw the region, then compute the center of mass \((\bar{x},\bar{y}).\) Use symmetry to help locate the center of mass whenever possible.

    20) [T] The region bounded by \(y=\cos(2x), \; x=−\frac{π}{4}\), and \(x=\frac{π}{4}\)

    Answer
    \(\left(0,\frac{π}{8}\right)\)

    21) [T] The region between \(y=2x^2, \; y=0, \; x=0,\) and \(x=1\)

    22) [T] The region between \(y=\frac{5}{4}x^2\) and \(y=5\)

    Answer
    \((0,3)\)

    23) [T] Region between \(y=\sqrt{x}, \; y=\ln x, \; x=1,\) and \(x=4\)

    24) [T] The region bounded by \(y=0\) and \(\dfrac{x^2}{4}+\dfrac{y^2}{9}=1\)

    Answer
    \(\left(0,\frac{4}{π}\right)\)

    25) [T] The region bounded by \(y=0, \; x=0,\) and \(\dfrac{x^2}{4}+\dfrac{y^2}{9}=1\)

    26) [T] The region bounded by \(y=x^2\) and \(y=x^4\) in the first quadrant

    Answer
    \(\left(\frac{5}{8},\, \frac{1}{3}\right)\)

    In exercises 27 - 31, use the theorem of Pappus to determine the volume of the shape.

    27) Rotating \(y=mx\) around the \(x\)-axis between \(x=0\) and \(x=1\)

    28) Rotating \(y=mx\) around the \(y\)-axis between \(x=0\) and \(x=1\)

    Answer
    \(V = \frac{mπ}{3}\) units³

    29) A general cone created by rotating a triangle with vertices \((0,0), \, (a,0),\) and \((0,b)\) around the \(y\)-axis. Does your answer agree with the volume of a cone?

    30) A general cylinder created by rotating a rectangle with vertices \((0,0), \, (a,0), \, (0,b),\) and \((a,b)\) around the \(y\)-axis. Does your answer agree with the volume of a cylinder?

    Answer
    \(V = πa^2b\) units³

    31) A sphere created by rotating a semicircle with radius \(a\) around the \(y\)-axis. Does your answer agree with the volume of a sphere?

    In exercises 32 - 36, use a calculator to draw the region enclosed by the curve. Find the area \(M\) and the centroid \((\bar{x},\bar{y})\) for the given shapes. Use symmetry to help locate the center of mass whenever possible.

    32) [T] Quarter-circle: \(y=\sqrt{1−x^2}, \; y=0\), and \(x=0\)

    Answer
    \(\left(\frac{4}{3π},\, \frac{4}{3π}\right)\)

    33) [T] Triangle: \(y=x, \; y=2−x\), and \(y=0\)

    34) [T] Lens: \(y=x^2\) and \(y=x\)

    Answer
    \(\left(\frac{1}{2},\, \frac{2}{5}\right)\)

    35) [T] Ring: \(y^2+x^2=1\) and \(y^2+x^2=4\)

    36) [T] Half-ring: \(y^2+x^2=1, \; y^2+x^2=4,\) and \(y=0\)

    Answer
    \(\left(0,\, \frac{28}{9π}\right)\)

    37) Find the generalized center of mass in the sliver between \(y=x^a\) and \(y=x^b\) with \(a>b\). Then, use the Pappus theorem to find the volume of the solid generated when revolving around the \(y\)-axis.

    38) Find the generalized center of mass between \(y=a^2−x^2, \; x=0\), and \(y=0\). Then, use the Pappus theorem to find the volume of the solid generated when revolving around the \(y\)-axis.

    Answer
    Center of mass: \(\left(\frac{a}{6},\,\frac{4a^2}{5}\right),\)
    Volume: \(\dfrac{2πa^4}{9}\) units³

    39) Find the generalized center of mass between \(y=b\sin(ax),\; x=0,\) and \(x=\dfrac{π}{a}.\) Then, use the Pappus theorem to find the volume of the solid generated when revolving around the \(y\)-axis.

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

    This figure is a torus. It has inner radius of b. Inside of the torus is a cross section that is a circle. The circle has radius a.

    Answer
    Volume: \(V = 2\pi^2a^2(b+a)\)

    41) Find the center of mass \((\bar{x},\bar{y})\) for a thin wire along the semicircle \(y=\sqrt{1−x^2}\) with unit mass. (Hint: Use the theorem of Pappus.)

    Contributors

    Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

     


    This page titled 6.6E: Exercises for Section 6.6 is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.