<ol class="i-list"><li><div class="list-item-paragraph">Find the exact value of $\int_{0}^{1}{\frac{1}{1+{{x}^{2}}}\,\mathrm{d}x}$.</div></li><li><div class="list-item-paragraph">The graph of $y\,=\,\frac{1}{1+{{x}^{2}}}$, for $0\le x\le 1$, is shown in the diagram. Rectangles, each of width $\frac{1}{n}$, are drawn under the curve. Show that the total area $A$ of all $n$ rectangles is given by $A\,=\,\frac{1}{n}\left\{ \frac{1}{1+{{\left( \frac{1}{n} \right)}^{2}}}+\frac{1}{1+{{\left( \frac{2}{n} \right)}^{2}}}+\frac{1}{1+{{\left( \frac{3}{n} \right)}^{2}}}+...+\frac{1}{2} \right\}$.</div><div class="list-item-paragraph">State the limit of $A$ as $n\to \infty $.</div><div class="list-item-image"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1760258639491-i19fo6oaj8.jpg" alt="List item image" /></div></li><li><div class="list-item-paragraph">It is given that $B\,=\,\frac{1}{n}\left\{ \frac{1}{1+{{\left( \frac{1}{n} \right)}^{4}}}+\frac{1}{1+{{\left( \frac{2}{n} \right)}^{4}}}+\frac{1}{1+{{\left( \frac{3}{n} \right)}^{4}}}+...+\frac{1}{2} \right\}$. Find using the aid of a calculator the limit of $B$ as $n\to \infty $ correct to $3$ significant figures.</div></li></ol>

Find the exact value of $\displaystyle\displaystyle \int_{0}^{1}{\frac{1}{1+{{x}^{2}}}\,\mathrm{d}x}$.

The graph of $y\,=\,\frac{1}{1+{{x}^{2}}}$, for $0\le x\le 1$, is shown in the diagram. Rectangles, each of width $\frac{1}{n}$, are drawn under the curve. Show that the total area $A$ of all $n$ rectangles is given by $A\,=\,\frac{1}{n}\left\{ \frac{1}{1+{{\left( \frac{1}{n} \right)}^{2}}}+\frac{1}{1+{{\left( \frac{2}{n} \right)}^{2}}}+\frac{1}{1+{{\left( \frac{3}{n} \right)}^{2}}}+...+\frac{1}{2} \right\}$.

State the limit of $A$ as $n\to \infty $.

It is given that $B\,=\,\frac{1}{n}\left\{ \frac{1}{1+{{\left( \frac{1}{n} \right)}^{4}}}+\frac{1}{1+{{\left( \frac{2}{n} \right)}^{4}}}+\frac{1}{1+{{\left( \frac{3}{n} \right)}^{4}}}+...+\frac{1}{2} \right\}$. Find using the aid of a calculator the limit of $B$ as $n\to \infty $ correct to $3$ significant figures.

N2001 P1 Q18 [Modified]

N2001 P1 Q18 [Modified]

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