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Math Help - Solids of Known Cross sections(volume)

  1. #1
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    Solids of Known Cross sections(volume)

    The base of a solid is bounded by y=x^3, y=0, and x=1. Find the volume of the solid for each of the following cross sections perpendicular to the y-axis.

    a)semicircles

    b)Squares

    -MY STEPS

    a) A =\frac{\pi r^2}{2}

    r = (\sqrt[3]{y/2})

    V = \pi /4 \int_{0}^{1} (\sqrt[3]{y})^2

    V=.4712319
    (Did I do this right?)

    b) A=D^2

    D=\sqrt[3]{y}
    V=\int_{0}^{1}(\sqrt[3]{y})^2
    V=.600000
    (did I do this right?)

    I don't think I did these right. Can someone help me out. Thanks.
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    South Coast of England
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    Volume problems

    Hello dandaman
    Quote Originally Posted by dandaman View Post
    The base of a solid is bounded by y=x^3, y=0, and x=1. Find the volume of the solid for each of the following cross sections perpendicular to the y-axis.

    a)semicircles

    b)Squares

    -MY STEPS

    a) A =\frac{\pi r^2}{2}

    r = (\sqrt[3]{y/2})

    V = \pi /4 \int_{0}^{1} (\sqrt[3]{y})^2

    V=.4712319
    (Did I do this right?)

    b) A=D^2

    D=\sqrt[3]{y}
    V=\int_{0}^{1}(\sqrt[3]{y})^2
    V=.600000
    (did I do this right?)

    I don't think I did these right. Can someone help me out. Thanks.
    y = x^3 \Rightarrow \frac{dy}{dx} = 3x^2 \Rightarrow dy = 3x^2dx

    (a) Take a cross-section at right-angles to the y-axis at the point (x, y) on y = x^3. Then the semi-circle has diameter (1-x), and area \pi(\tfrac{1}{2}(1-x))^2 = \frac{\pi}{4}(1-x)^2. So the volume is:

    \int_0^1\frac{\pi}{4}(1-x)^2dy=\int_0^1\frac{\pi}{4}(1-x)^23x^2\,dx

    (b) With a square cross-section, the area of cross-section is (1-x)^2. So the volume is

    \int_0^1(1-x)^23x^2\,dx

    Can you complete them now?

    Grandad
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