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Integration to Form a Solid - Using Shells?

  1. Jan 25, 2009 #1


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    Integration to Form a Solid -- Using Shells?

    1. The problem statement, all variables and given/known data

    Use the shell method to find the volumes of solids generated by revolving the shaded regions
    about the indicated axes. The graph shows the equations:

    [tex]x = \dfrac{y^4}{4}-\dfrac{y^2}{2}[/tex]
    [tex]x = \dfrac{y^2}{2}[/tex]

    a) The x-axis
    b) The line y = 2
    c) The line y = 5
    d) The line y = -5/8

    2. Relevant equations

    Shell method: [tex]V = 2\pi\int_a^b \! y * f(y) \, dy [/tex]
    3. The attempt at a solution
    I attempted to use the shell method to find the volume of the formed "bowl" but I get a negative number:

    [tex]V = 2\pi\int_0^2 \! y * (\dfrac{y^4}{4}-\dfrac{y^2}{2} - \dfrac{y^2}{2}) \, dy = 2\pi\int_a^b \! y * (\dfrac{y^4}{4}-y^2) \, dy = 2\pi\int_a^b \! (\dfrac{y^5}{4}-y^3) \, dy = 2\pi[\dfrac{y^6}{24} - \dfrac{y^4}{4}]^2_0 = 2\pi[\dfrac{64}{24} - \dfrac{16}{4}][/tex]

    Where am I going wrong?

    - x^2
  2. jcsd
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