N2015 P1 Q3

Given that \(\mathrm{f}\) is a continuous function, explain, with the aid of a sketch, why the value of \({\displaystyle \lim_{n\to\infty} \frac{1}{n}\left\{\mathrm{f}\left(\frac{1}{n}\right)+\mathrm{f}\left(\frac{2}{n}\right)+\ldots+\mathrm{f}\left(\frac{n}{n}\right)\right\}}\) is also \({\displaystyle \int_0^1 \mathrm{f}(x)\mathrm{d}x}\).[2]

Hence evaluate \({\displaystyle \lim_{n\to\infty} \frac{1}{n}\left(\frac{\sqrt[3]{1}+\sqrt[3]{2}+\ldots+\sqrt[3]{n}}{\sqrt[3]{n}}\right).}\)[3]