Area of the region bounded by curve \({\mathrm{y}}^{2}=4x\) and the X -axis between \(x=0\) and \(x=1\) is :

\(y=\sqrt{4x}=2\sqrt{x}\)

The required area can be calculated as:

\(\begin{matrix}A & = & {\int }_{0}^{1}2\sqrt{x}dx \\ & = & 2{\int }_{0}^{1}{x}^{1/2}dx \\ & = & 2{\left[\frac{{x}^{1/2+1}}{1/2+1}\right]}_{0}^{1} \\ & = & 2{\left[\frac{2}{3}{x}^{3/2}\right]}_{0}^{1} \\ & = & \frac{4}{3}\left({1}^{3/2}-{0}^{3/2}\right) \\ & = & \frac{4}{3}(1-0) \\ & = & \frac{4}{3}\end{matrix}\)