Evaluate the double integral; $I={\displaystyle \underset{0}{\overset{3}{}}}{\displaystyle \underset{1}{\overset{2}{}}}{x}^{2}ydydx$

Note: Write your final answer as a fraction, not a decimal.

$2.$ Evaluate the volume of the solid bounded above by the surface: $z=2xsin(y),$ and below by the region in the $xy-$plane enclosed by the curves: $y=0,y=x^2,x=0$ and $x=1.$

$3.$ Find the centre of mass of a lamina with density $(x,y)=xy$ over the region bounded by: $y=0,y=x^2$ from $x=0$ to $x=2.$

Note: Write your final answers as fractions, not decin