Official solution
Area of the shaded region
\[ = 2\int_0^1 {\left( {{{{y^2} + 3} \over 4} - {{{y^2} + 1} \over 2}} \right)dy} \]
\[ = 2\int_0^1 {\left( {{1 \over 4} - {{{y^2}} \over 4}} \right)dy} \]
\[ = 2\left[ {{1 \over 4} - {1 \over {12}}} \right] = {1 \over 3}\]