<p>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 $A$.</p>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1760850560508-weww0d34ntd.png" width="400" class="aligncenter" /></figure>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that the total area of $n$ rectangles is given by $\frac{1}{n\sqrt{n}}\sum\limits_{r=0}^{n-1}{\sqrt{n-r}}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Without using a calculator, evaluate $\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n\sqrt{n}}\sum\limits_{r=0}^{n-1}{\sqrt{n-r}}$.</div><p style="text-align: right;">[3]</p></li></ol>

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 $A$.

Show that the total area of $n$ rectangles is given by $\frac{1}{n\sqrt{n}}\sum\limits_{r=0}^{n-1}{\sqrt{n-r}}$.

Without using a calculator, evaluate $\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n\sqrt{n}}\sum\limits_{r=0}^{n-1}{\sqrt{n-r}}$.

2024 HCI Promo Q3

2024 HCI Promo Q3

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