2023 HCI Promo Q4

The shaded region \({R}\) is bounded by the \({x}\)-axis, the \({y}\)-axis, the line \(y=1\) and the curve \({x={{y}^{2}}+1}\). There are \({n}\) rectangles of equal width drawn as shown in the diagram below. The total area of all the n rectangles, \({{A}_{n}}\) approximates the area of the shaded region \({R}\).

It is given that \({\mathop{\sum }_{r=1}^{n}{{r}^{2}}=\frac{n\left( n+1 \right)\left( 2n+1 \right)}{6}}\).

1. Show that \({{{A}_{n}}=1+\frac{\left( n-1 \right)\left( 2n-1 \right)}{6{{n}^{2}}}}\).[3]
2. Find the exact value of \({\mathop{\lim}\limits_{n\to \infty }\,{{A}_{n}}}\), showing your working.[3]