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Mathematics LibreTexts

1.5E: Exercises

  • Page ID
    18543
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    Moments and Centers of Mass

    For the following exercises, calculate the center of mass for the collection of masses given.

    Exercise \(\PageIndex{1}\)

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

    Answer

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    Exercise \(\PageIndex{2}\)

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

    Answer

    \(\displaystyle \frac{5}{4}\)

    Exercise \(\PageIndex{3}\)

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

    Answer

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    Exercise \(\PageIndex{4}\)

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

    Answer

    \(\displaystyle (\frac{2}{3},\frac{2}{3})\)

    Exercise \(\PageIndex{5}\)

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

    Answer

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    Exercise \(\PageIndex{6}\)

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

    Answer

    \(\displaystyle (\frac{7}{4},\frac{3}{2})\)

    For the following exercises, compute the center of mass x–.

    Exercise \(\PageIndex{7}\)

    \(\displaystyle ρ=1\) for \(\displaystyle x∈(−1,3)\)

    Answer

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    Exercise \(\PageIndex{8}\)

    \(\displaystyle ρ=x^2\) for \(\displaystyle x∈(0,L)\)

    Answer

    \(\displaystyle \frac{3L}{4}\)

    Exercise \(\PageIndex{9}\)

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

    Answer

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    Exercise \(\PageIndex{10}\)

    \(\displaystyle ρ=sinx\) for \(\displaystyle x∈(0,π)\)

    Answer

    \(\displaystyle \frac{π}{2}\)

    Exercise \(\PageIndex{11}\)

    \(\displaystyle ρ=cosx\) for \(\displaystyle x∈(0,\frac{π}{2})\)

    Answer

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    Exercise \(\PageIndex{12}\)

    \(\displaystyle ρ=e^x\) for \(\displaystyle x∈(0,2)\)

    Answer

    \(\displaystyle \frac{e^2+1}{e^2−1}\)

    Exercise \(\PageIndex{13}\)

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

    Answer

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    Exercise \(\PageIndex{14}\)

    \(\displaystyle ρ=xsinx\) for \(\displaystyle x∈(0,π)\)

    Answer

    \(\displaystyle \frac{π^2−4}{π}\)

    Exercise \(\PageIndex{15}\)

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

    Answer

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    Exercise \(\PageIndex{16}\)

    \(\displaystyle ρ=lnx\) for \(\displaystyle x∈(1,e)\)

    Answer

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

    For the following exercises, compute the center of mass \(\displaystyle (\bar{x},\bar{y})\). Use symmetry to help locate the center of mass whenever possible.

    Exercise \(\PageIndex{17}\)

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

    Answer

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    Exercise \(\PageIndex{18}\)

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

    Answer

    \(\displaystyle (\frac{a}{3},\frac{b}{3})\)

    Exercise \(\PageIndex{19}\)

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

    Answer

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

    Exercise \(\PageIndex{20}\)

    The region bounded by \(\displaystyle y=cos(2x), x=−\frac{π}{4}\), and \(\displaystyle x=\frac{π}{4}\)

    Answer

    \(\displaystyle (0,\frac{π}{8})\)

    Exercise \(\PageIndex{21}\)

    The region between \(\displaystyle y=2x^2, y=0, x=0,\) and \(\displaystyle x=1\)

    Answer

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    Exercise \(\PageIndex{22}\)

    The region between \(\displaystyle y=\frac{5}{4}x^2\) and \(\displaystyle y=5\)

    Answer

    \(\displaystyle (0,3)\)

    Exercise \(\PageIndex{23}\)

    Region between \(\displaystyle y=\sqrt{x}, y=ln(x), x=1,\) and \(\displaystyle x=4\)

    Answer

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    Exercise \(\PageIndex{24}\)

    The region bounded by \(\displaystyle y=0, \frac{x^2}{4}+\frac{y^2}{9}=1\)

    Answer

    \(\displaystyle (0,\frac{4}{π})\)

    Exercise \(\PageIndex{25}\)

    The region bounded by \(\displaystyle y=0, x=0,\) and \(\displaystyle \frac{x^2}{4}+\frac{y^2}{9}=1\)

    Answer

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    Exercise \(\PageIndex{26}\)

    The region bounded by \(\displaystyle y=x^2\) and \(\displaystyle y=x^4\) in the first quadrant

    Answer

    \(\displaystyle (\frac{5}{8},\frac{1}{3})\)

    For the following exercises, use the theorem of Pappus to determine the volume of the shape.

    Exercise \(\PageIndex{27}\)

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

    Answer

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    Exercise \(\PageIndex{28}\)

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

    Answer

    \(\displaystyle \frac{mπ}{3}\)

    Exercise \(\PageIndex{29}\)

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

    Answer

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    Exercise \(\PageIndex{30}\)

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

    Answer

    \(\displaystyle πa^2b\)

    Exercise \(\PageIndex{31}\)

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

    Answer

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

    Exercise \(\PageIndex{32}\)

    Quarter-circle: \(\displaystyle y=\sqrt{1−x^2}, y=0\), and \(\displaystyle x=0\)

    Answer

    \(\displaystyle (\frac{4}{3π},\frac{4}{3π})\)

    Exercise \(\PageIndex{33}\)

    Triangle: \(\displaystyle y=x, y=2−x\), and \(\displaystyle y=0\)

    Answer

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    Exercise \(\PageIndex{34}\)

    Lens: \(\displaystyle y=x^2\) and \(\displaystyle y=x\)

    Answer

    \(\displaystyle (\frac{1}{2},\frac{2}{5})\)

    Exercise \(\PageIndex{35}\)

    Ring: \(\displaystyle y^2+x^2=1\) and \(\displaystyle y^2+x^2=4\)

    Answer

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    Exercise \(\PageIndex{36}\)

    Half-ring: \(\displaystyle y^2+x^2=1, y^2+x^2=4,\) and \(\displaystyle y=0\)

    Answer

    \(\displaystyle (0,\frac{28}{9π})\)

    Exercise \(\PageIndex{37}\)

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

    Answer

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    Exercise \(\PageIndex{38}\)

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

    Answer

    Center of mass: \(\displaystyle (\frac{a}{6},\frac{4a^2}{5}),\) volume: \(\displaystyle \frac{2πa^4}{9}\)

    Exercise \(\PageIndex{39}\)

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

    Answer

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    Exercise \(\PageIndex{40}\)

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

    Answer

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