Problem

The diagram shows the curve with equation $y = \frac{1}{3}x^3 + x$ for $0 \leq x \leq 1$, 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

$$\int_0^1 \bigg(\frac{1}{3}x^3 + x\bigg)\,\mathrm{d}x < U_n,$$

where

$$U_n = \frac{1}{12}\bigg(1 + \frac{1}{n}\bigg) \bigg(7 + \frac{1}{n}\bigg).$$

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

$$\int_0^1 \bigg(\frac{1}{3}x^3 + x\bigg)\,\mathrm{d}x.$$

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