Question $2.(10+5$ points$)$

$(a)$ Verify Green's Theorem for $o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}{x}^2{y}^{2}dx+xydy,$ where $C$ consists

$of$ the arc $of$ parabola $y=x^2$ from $(0,0)to(1,1),$ followed $by$ the line

segments from $(1,1)to(0,1),$ and from $(0,1)to(0,0).$

$(b)$ Let a lamina lying $inxy-$plane $is$ occupying a region $D$ which

$is$ bounded $by$ a simple closed path $C.$ Let $Abe$ the area $ofD.$ Using

Green's Theorem show that $if$ the lamina has constant density then

the centre $of$ mass $of$ the lamina has coordinates $(\stackrel{}{x}\frac{,}{b}ar(y)),$ where

$\stackrel{}{x}=\frac{1}{2A}o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}{x}^{2}dy\frac{,}{b}ar(y)=\frac{-1}{2A}o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}{y}^{2}dx$