Recognise a right-endpoint sum and a cube-root limit

(i) For continuous $f$, explain with a sketch why

$$\lim_{n\to\infty}\frac1n\left[f\left(\frac1n\right)+f\left(\frac2n\right)+\cdots+f\left(\frac nn\right)\right]=\int_0^1f(x)\,dx.$$

(ii) Hence evaluate $\displaystyle\lim_{n\to\infty}\frac1n\left(\frac{\sqrt[3]1+\sqrt[3]2+\cdots+\sqrt[3]n}{\sqrt[3]n}\right)$.3 marks