The area bounded by the curve $$y = \sqrt x ,$$  the line $$2y+3=x$$   and the $$x$$-axis in the first quadrant is :

Solution :

$$\eqalign{ & {\text{Solving }}y = \sqrt x {\text{ and }}y = \frac{x}{2} - \frac{3}{2},\,\left( {x - 1} \right)\left( {x - 9} \right) = 0 \cr & \therefore x = 9,{\text{ }}1.{\text{ But }}y{\text{ is positive, so }}x \ne 1. \cr & {\text{Hence, }}x = 9. \cr & \therefore {\text{Area}} = \int_0^9 {\sqrt x \,dx - \int_3^9 {\left( {\frac{1}{2}x - \frac{3}{2}} \right)} } dx \cr} $$