$(a)$ Let $be$ the region $in{R}^2$ bounded $byx=1,x=2,y=x$ and $y=x^3.$

$i.$ Sketch $.$

$ii.$ Determine ${}_{}\sqrt[\phantom{2}]{xy}dxdy.$

$(b)$ Determine the following integral $by$ changing $to$ polar coordinates

${\displaystyle \underset{-1}{\overset{1}{}}}{\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{1-y^2}}{}}}{e}^{-(x^2+y^2)}dxdy$

$[12$ marks$]$

$(c)$ Let $Cbe$ the line segment $in{R}^2$ from $(1,1)to(2,5).$ Determine the line

integral

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}\frac{y}{x}dl$