Parabola, tangents and superficies

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 $M + N$ ?