From four rectangles to an exact area

The diagram shows $y=x^2$ between $x=0$ and $x=1$, with four rectangles of equal width.

(i) Show that their total area is $\frac1{4^3}\sum_{r=1}^4r^2$.

(ii) If $n$ rectangles are used instead, deduce an expression for their total area.

(iii) Show that $\lim_{n\to\infty}\frac1{n^3}\sum_{r=1}^nr^2=\frac13$. What does this limit represent?

(iv) Give a reason why $\frac1{n^3}\sum_{r=1}^nr^2>\frac13$. You may use $\sum_{r=1}^nr^2=\frac n6(n+1)(2n+1)$ without proof.