Let \(y=\mathrm{e}^{x}\). Find the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant 2\).

Show that the mean value of \(x\) with respect to \(y\) over the interval \(1 \leqslant y \leqslant \mathrm{e}^{2}\) is \(\frac{\mathrm{e}^{2}+1}{\mathrm{e}^{2}-1}\).