All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)

Width of each rectangle $= \frac{1}{n}$
Total area of rectangles $= \frac{1}{n} \sum_{r=1}^n f(\frac{r}{n})$

When $n \rightarrow \infty$, we have the exact area instead.

$\therefore, \lim_{n\to\infty} \frac{1}{n} \sum_{r=1}^n f(\frac{r}{n}) = \int_0^1 f(x) dx$

(ii)
$\lim_{n\to\infty} \frac{1}{n} [(\frac{1}{n})^{\frac{1}{3}} + (\frac{2}{n})^{\frac{1}{3}} + \ldots + (\frac{n}{n})^{\frac{1}{3}}]$

$= \lim_{n\to\infty} \frac{1}{n} \sum_{r=1}^n f(\frac{r}{n})$ where $f(x) = x^{\frac{1}{3}}$

$= \int_0^1 f(x) dx$

$= \frac{3x^{\frac{4}{3}}}{4} \bigl|_0^1$

$= \frac{3}{4}$