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Math Help - finding the volume

  1. #1
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    finding the volume

    Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x axis. Sketch the region and a typical shell

    x + y = 3, x = 4 - (y - 1)^2
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  2. #2
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    Quote Originally Posted by gracy View Post
    Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x axis. Sketch the region and a typical shell

    x + y = 3, x = 4 - (y - 1)^2
    For the sketch,
    -----x +y = 3 is a straight line passing through (0,3) and (3,0).
    -----x = 4 -(y-1)^2 is a horizontal parabola that opens to the left whose vertex is at (4,1), and whose y-intercepts are (0,3) and (0,-1), and whose x-intercept is (3,0).

    If by shell method, rotating about the x-axis,
    dV = [2pi(y)(x2 -x1)](dy)-----where x2 is from the parabola and x1 is from the line.
    dV = 2pi(y)[(4 -(y-1)^2) -(3-y)]dy
    dV = 2pi(y)[(4 -(y^2 -2y +1) -3 +y]dy
    dV = 2pi(y)[4 -y^2 +2y -1 -3 +y]dy
    dV = 2pi(y)[-y^2 +3y]dy
    dV = 2pi[3y^2 -y^3]dy ---------------**

    The boundaries of dy are from y=0 up to y=3.

    So,
    V = (2pi)INT.(0-->3)[3y^2 -y^3]dy
    V = (2pi)[y^3 -(1/4)y^4] |(0-->3)
    V = (2pi){[(3^3) -(1/4)(3^4)] -[0]}
    V = (2pi){(27) -(81/4)]
    V = (2pi)(27)[1 -3/4]
    V = (54pi)[1/4]
    V = 13.5pi cu.units ----------------answer.
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