Problem

The diagram shows the curve with equation $y=2x-x^2$ for $0\le x\le1$, together with a set of $n$ rectangles of width $\frac{1}{n}$.

(a) By considering the sum of the areas of these rectangles, show that

$$\int_0^1(2x-x^2)\,\mathrm{d}x<U_n,$$

where

$$U_n=\left(1+\frac{1}{n}\right)\left(\frac{2}{3}-\frac{1}{6n}\right).$$

(b) Use a similar method to find, in terms of $n$, a lower bound $L_n$ for

$$\int_0^1(2x-x^2)\,\mathrm{d}x.$$

(c) Show that

$$\lim_{n\to\infty}(U_n-L_n)=0.$$