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calculus

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Find the volume of the solid generated by revolving about line x = -4 the region bounded by x = y - y^2 and x = y^2 - 3

  • calculus - ,

    The curves intersect at (-2,-1) and (-3/4,3/2)

    So, we want to sum up the washers with inner radius r and outer radius R

    Int pi*(R^2 - r^2)[-1,3/2] dy

    where
    R = y - y^2 + 4
    r = y^2 - 3 + 4 = y^2 + 1

    R^2 - r^2
    = (y^2 - y^3 + 4y - y^3 + y^4 - 4y^2 + 4y - 4y^2 + 16) - (y^4 + 2y^2 + 1)
    = -2y^3 - 9y^2 + 8y + 15

    V = pi*(-1/2 y^4 - 3y^3 + 4y^2 + 15y)[-1,3/2]
    = 1085pi/96

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