Area Under The Curves

Let the locus of the centre \((\alpha, \beta), \beta>0\), of the circle which touches the circle \(x^{2}+(y-1)^{2}=1\) externally and also touches the \(x\)-axis be \(\mathrm{L}\). Then the area bounded by \(\mathrm{L}\) and the line \(y=4\) is: