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Math Help - Volume of object, shell method

  1. #1
    Member nautica17's Avatar
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    Volume of object, shell method

    I did this problem and I'm not sure if it's correct since the answer in the back of the book is different.. yet my own work and Wolfram Alpha says differently. Could someone verify this for me?

    Question: Use cylindrical shells to to find the volume of the solid that is generated when the region under the curve y = x^3 - 3x^2 + 2x over [0,1] is revolved about the y-axis.

    My work:
    integral from 0 to 1 of 2pi * (x^3 - 3x^2 + 2x) dx
    = 2pi *[ (x^4)/4 - x^3 + x^2 ] evaluated from 0 to 1
    = 2pi ( [(1^4)/4 - 1^3 + 1^2] - 0 )
    = 2pi / 4 = pi / 2 <-final answer

    My book however says that the answer is 7pi / 30. I'm a little frustrated with this one. Where did I go wrong?
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  2. #2
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    Quote Originally Posted by nautica17 View Post
    I did this problem and I'm not sure if it's correct since the answer in the back of the book is different.. yet my own work and Wolfram Alpha says differently. Could someone verify this for me?

    Question: Use cylindrical shells to to find the volume of the solid that is generated when the region under the curve y = x^3 - 3x^2 + 2x over [0,1] is revolved about the y-axis.

    My work:
    integral from 0 to 1 of 2pi * (x^3 - 3x^2 + 2x) dx
    = 2pi *[ (x^4)/4 - x^3 + x^2 ] evaluated from 0 to 1
    = 2pi ( [(1^4)/4 - 1^3 + 1^2] - 0 )
    = 2pi / 4 = pi / 2 <-final answer

    My book however says that the answer is 7pi / 30. I'm a little frustrated with this one. Where did I go wrong?
    you forgot the radius of rotation, x ...

    \displaystyle V = 2\pi \int_0^1 x(x^3-3x^2+2x) \, dx
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