The diagram shows the curve with equation \(y=2^{x}\) for \(0 \leqslant x \leqslant 1\), 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 \(\int_{0}^{1} 2^{x} \mathrm{~d} x\lt U_{N}\), where
\(U_{N}=\frac{2^{\frac{1}{N}}}{N\left(2^{\frac{1}{N}}-1\right)} .\)
(b) Use a similar method to find, in terms of \(N\), a lower bound \(L_{N}\) for \(\int_{0}^{1} 2^{x} \mathrm{~d} x\).

(c) Find the least value of \(N\) such that \(U_{N}-L_{N}\lt 10^{-4}\).