Question

Use Green's Theorem to evaluate \[\int\limits_{C}^{}xydx+x ^{2}y ^{3}dy\] where C is the triangle with vertices (0,0), (1,0), and (1,2).

Answer 1

Take your partial derivatives:
\[N_x=\frac{\partial (x^2y^3)}{\partial x}=\cdots\\
M_y=\frac{\partial (xy)}{\partial y}=\cdots\]
Then by Green's theorem, you have
\[\int_C M~dx+N~dy=\int\int_D(N_x-M_y)~dA\]
where \(D\) is the triangular region defined by
\[D:=\left\{(x,y)~:~0\le x\le1,~0\le y\le2x\right\}\]