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Math Help - integrating with respect to y / regions between curves

  1. #1
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    integrating with respect to y / regions between curves

    Find the region R bounded by the graphs y = x^3 and  y = x+ 6

    The left curve y=x+6 becomes x= y-6
    The right curve y=x^3 becomes  x=y^{1/3}



    How do I find the intersection point of the curve and the root which will be the pper and lower boundaries. The book says I can use synthetic division to find the roots.
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  2. #2
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    Re: integrating with respect to y / regions between curves

    The two curves will intersect when x^3 = x + 6. Thus, you must solve x^3 - x - 6 = 0. It is relatively easy to tell that 2 is a root of p(x) = x^3 - x - 6. Thus, by synthetic or long division we can factor p(x) as (x - 2)(x^2 + 2x + 3). Since x^2 + 2x + 3 does not factor, we have that p(x) has only one real root, which means that y = x^3 intersects y = x + 6 at exactly one point, which means that there is no region R bounded by the graphs.
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