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Homework Help: Solid of Rotation:I Need Verification

  1. May 7, 2008 #1
    1. The problem statement, all variables and given/known data
    Determine the volume of a sold obtained by rotating the region bounded by y=x^2-2x and y=x about the line y =4



    2. Relevant equations



    3. The attempt at a solution
    Let:
    y1 = x
    y2 = x^2 - 2x.
    v be the volume.

    y1 and y2 meet where:
    x^2 - 2x = x
    x^2 - 3x = 0
    x(x - 3) = 0
    x = 0, x = 3.

    The straight line is closer to y = 4 than the parabola between x = 0 and x = 3.

    v = pi int(0, 3)[ (y2 - 4)^2 - (y1 - 4)^2 ] dx
    = pi int(0, 3) [ (x^2 - 2x - 4)^2 - (x - 4)^2 ] dx
    = pi int(0, 3) [ x^4 + 4x^2 + 16 - 4x^3 + 16x - 8x^2 - x^2 - 16 + 8x ] dx
    = pi int(0, 3) [ x^4 - 4x^3 - 5x^2 + 24x ] dx
    = pi [ x^5 / 5 - x^4 - 5x^3 / 3 + 12x^2 ](0, 3)
    = pi [ 3^5 / 5 - 3^4 - 5 * 3^3 / 3 + 12 * 3^2 ]
    = 30.6 pi.

    IS THIS RIGHT?
     
  2. jcsd
  3. May 7, 2008 #2

    Defennder

    User Avatar
    Homework Helper

    I got the same answer.
     
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