Problem

The diagram shows the curve with equation $y = \frac{1}{2}(3^x)$ for $0 \le x \le 1$, together with a set of $N$ rectangles each of width $\frac{1}{N}$.

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

$$\int_0^1 \frac{1}{2}(3^x)\,\mathrm{d}x < U_N,$$

where

$$U_N = \frac{3^{\frac{1}{N}}}{N\big(3^{\frac{1}{N}} - 1\big)}.$$

(b) Use a similar method to find, in terms of $N$, a lower bound $L_N$ for $\int_0^1 \frac{1}{2}(3^x)\,\mathrm{d}x$.

(c) By simplifying $U_N - L_N$, show that $\lim_{N \to \infty}(U_N - L_N) = 0$.