Parabola, tangents and superficies

Geometry Level pending

Consider a parabola \(P\) given by \( (x, x^2) \), with \(x \in \mathbb{R} \), and \((a_n)_{n \ge 0}\) a sequence given by \(a_n = n^\beta\), with \(\beta > 0 \in \mathbb{R}\). Consider the superficie \(S\) given by the region at the right side of the vertical axis (\( x=0\)), below the parabola \(P\) and above the lines tangents to the parabola \(P\) on the points \((a_n, a_n^2)\).

There is a interval \( 0 < \beta < \dfrac{M}{N}\) which makes the area of \(S\) be finite. What is the value of \(M + N\)?

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