A curve \(C\) has parametric equations \(x=\mathrm{e}^{t} \cos t, y=\mathrm{e}^{t} \sin t\), for \(0 \leqslant t \leqslant \pi\). Find the arc \(l=\)

Find the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.