Verify Green’s theorem in the plane for ∫c{(xy + y2)dx + x2dy} where C is the closed curve of the region bounded by y = x and y = x2.

$Now,\ we\ have\ to\ find\ the\ intersection\ of\ y=x\ and\ y=x^2.$

$RHS = x = x^2 $

$ x - x^2 = 0$

$ \Rightarrow x (1-x)= 0$

$i.e.\ x = 0,1$

$Hence,\ (0,0)\ and\ (1,1)\ will\ be\ the\ points\ of\ intersections.$

$According\ to\ Green's\ theorem,$

$\begin{aligned} \int_{c} M\ dx …