Evaluate the integral

${}_{R}2xcos(y^3)dydx$

where $R=\{\begin{array}{cc}(x,& y)in{R}^2|0y1\end{array}$ and $\{$:$0x3y\}.$

Let $Rbe$ the region $in$ the $xy-$plane given $by$

$R=\{(x,y)in{R}^2$:$y0$ and $4x^2+y^{2}9\}$

$(a)$ Sketch the region $R,$ and write down the corresponding region $Sof$ the $r-$plane, where

$(x,y)=(rcos,rsin).$

$(b)$ Evaluate the double integral

${}_R\frac{2y}{x^2+y^2}dxdy$