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Math Help - Volume of revolution

  1. #1
    MHF Contributor alexmahone's Avatar
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    Volume of revolution

    A cube C of edge length 1 is rotated around a line passing through two opposite vertices, thereby sweeping out a solid S of revolution. Find the volume of S.

    Any hints would be much appreciated.
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  2. #2
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    Quote Originally Posted by alexmahone View Post
    A cube C of edge length 1 is rotated around a line passing through two opposite vertices, thereby sweeping out a solid S of revolution. Find the volume of S.

    Any hints would be much appreciated.
    1. Draw a sketch. Basically you are dealing with a rectangle whose length is l=\sqrt{2} and whose width is w = 1.
    Split this rectangle into two congruent right triangles.

    2. By rotating the rectangle about it's diaogonal you'll get a solid composed of two cones and two frustrums of cones.

    The height of the solid is H=\sqrt{3}.
    The base radius of the two cones is: r_c=\dfrac{1 \cdot \sqrt{2}}{\sqrt{3}}=\dfrac13 \cdot \sqrt{6}
    The height of the two cones is: h_c=\dfrac13 \cdot \sqrt{3}
    The second radius of the two frustrums of cones is: r_{f2}=\dfrac14 \cdot \sqrt{6}
    The height of the two frustrums of cones is: h_f=\dfrac16 \cdot \sqrt{3}.

    3. To calculate the complete volume of the solid use the formulas of the volume of a cone: V_c = \dfrac13 \cdot \pi \cdot r_c^2 \cdot h_c

    and the formula of the volume of a drustrum of a cone: V_f=\dfrac13 \cdot h_f \left(r_c^2+r_c \cdot r_{f2}+r_{f2}^2  \right)
    Attached Thumbnails Attached Thumbnails Volume of revolution-rotatwuerfel.png  
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