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Math Help - Volumes by Cylinderical Shells

  1. #1
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    Volumes by Cylinderical Shells

    Find the volumes of the solids generated by revolving the regions about the given axes.

    The region bounded by y= square root x y= (x^2)/8
    a.) the x-axis
    b.) the y-axis
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  2. #2
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    points of intersection

    \sqrt{x} = \frac{x^2}{8}

    x = 0 is the obvious solution

    8 = x^{\frac{3}{2}}

    x = 4

    about the x-axis, use washers ...

    V = \pi \int_0^4 (\sqrt{x})^2 - \left(\frac{x^2}{8}\right)^2 \, dx

    ... or cylindrical shells

    V = 2\pi \int_0^2 y(\sqrt{8y} - y^2) \, dy


    about the y-axis, use cylindrical shells ...

    V = 2\pi \int_0^4 x\left(\sqrt{x} - \frac{x^2}{8}\right) \, dx

    ... or washers

    V = \pi \int_0^2 (\sqrt{8y})^2 - (y^2)^2 \, dy
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