By considering a Riemann sum with $n$ rectangles of equal width:

1. Show that

   $${\int }_0^{b}xdx\approx \frac{b^2}{n^2}\sum _{k=1}^n(k-1)$$

2. Hence, show that

   $${\int }_0^{b}xdx\approx \frac{b^2}{2}(1-\frac{1}{n})$$

3. Hence, show that

   $${\int }_0^{b}xdx=\frac{b^2}{2}$$

4. By sketching

   $$y=x,0\le x\le b$$

   explain why the result in part (c) is to be expected.