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.