Greene's Theorem On Triangular Region

Question

$\int_C xy^2\mathrm{d}x + 2x^2y\mathrm{d}y , $ C is the triangle with vertices (0,0), (2,2),(2,4) .

My attempt : I drew the region And I'm taking orientation counterclockwise wise but now I'm not getting how to parametrize the region because I know the equation of OA and OB. But how to solve it...

Please help me. Thank you !!

Answer 1

$$P=xy^2,~Q=2x^2y$$

$$\partial_xQ-\partial_yP=4xy-2xy=2xy$$

By Green's theorem, the integral equals:

$$I=\int_0^2\int_x^{2x}~2xy~~ dydx$$

You can proceed from here.