The area (in sq. units) of the region $$\left\{ {\left( {x,\,y} \right):x \geqslant 0,\,x + y \leqslant 3,\,{x^2} \leqslant 4y\,\,{\text{and}}\,y \leqslant 1 + \sqrt x } \right\}$$           is :

Solution :

Area of shaded region :
$$\eqalign{ & = \int\limits_0^1 {\left( {1 + \sqrt x } \right)dx} + \int\limits_1^2 {\left( {3 - x} \right)dx} - \int\limits_0^2 {\frac{{{x^2}}}{4}dx} \cr & = \left. x \right]_0^1 + \left. {\frac{{{x^{\frac{3}{2}}}}}{{\frac{3}{2}}}} \right]_0^1 + \left. {3x} \right]_1^2 - \left. {\frac{{{x^2}}}{2}} \right]_1^2 - \left. {\frac{{{x^3}}}{{12}}} \right]_0^2 \cr & = \frac{5}{2}\,{\text{sq}}{\text{. units}} \cr} $$