Official solution
Area of shaded region
= \[\int\limits_0^1 {\left( {1 + \sqrt x } \right)dx} + \int\limits_1^2 {\left( {3 - x} \right)dx} - \int\limits_0^2 {{{{x^2}} \over 4}dx} \]
= \[\left[ x \right]_0^1 + \left[ {{{{x^{{3 \over 2}}}} \over {{3 \over 2}}}} \right]_0^1\] + \[3\left[ x \right]_1^2 - \left[ {{{{x^2}} \over 2}} \right]_1^2 - \left[ {{{{x^3}} \over {12}}} \right]_0^2\]
= \[{5 \over 2}\] sq. unit