N2001 P1 Q18 [Modified]

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

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.