2024 HCI Promo Q3

The diagram below shows a curve with equation \(y=\sqrt{\left( 1-x \right)}\). Let \(A\) be the area bounded by the curve and the axes. A total of \(n\) rectangles, each of equal width, are constructed to approximate the value of \(A\).

1. Show that the total area of \(n\) rectangles is given by \(\frac{1}{n\sqrt{n}}\sum\limits_{r=0}^{n-1}{\sqrt{n-r}}\).[2]
2. Without using a calculator, evaluate \(\mathop{\lim}\limits_{n\to \infty }\,\frac{1}{n\sqrt{n}}\sum\limits_{r=0}^{n-1}{\sqrt{n-r}}\).[3]