The area bounded by the curves $$y = \sqrt x ,\,2y + 3 = x$$    and $$x$$-axis in the 1^st quadrant is-

Solution :
Given equation of the curves are for $$y = \sqrt x $$   and $$x = 2y + 3$$   in the first quadrant.
On solving both the equation for $$y$$, we get
$$\eqalign{ & y = \sqrt {2y + 3} \cr & \Rightarrow {y^2} = 2y + 3 \cr & \Rightarrow {y^2} - 2y - 3 = 0 \cr & \Rightarrow {y^2} - 3y + y - 3 = 0 \cr & \Rightarrow y\left( {y - 3} \right) + 1\left( {y - 3} \right) = 0 \cr & \Rightarrow \left( {y + 1} \right)\left( {y - 3} \right) = 0 \cr & \Rightarrow y = - 1,\,\,3 \cr} $$
∴ Required area of shaded region,
$$\eqalign{ & A = \int_0^3 {\left( {2y + 3 - {y^2}} \right)dy} = \left[ {\frac{{2{y^2}}}{2} + 3y - \frac{{{y^3}}}{3}} \right]_0^3 \cr & = \left[ {\frac{{18}}{2} + 9 - 9 - 0} \right] \cr & = 9\,{\text{sq}}{\text{. units}} \cr} $$