How to solve this Green's theorem Question?

Question

I have been given the question by my university as an Assignment to solve. I have solved but when I shared with my fellows they are saying the answer of this question is zero, but I am getting the other answer. I have tried all the ways, I am so confused that how the answer is zero? Kindly anyone help me in this.

QUESTION:

MY TRY:

Kindly anyone help me in this.

Answer 1

You wrote the formula wrong. $$\oint_{C}(Ldx+Mdy)=\iint_{D}\left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)dxdy$$

$$\oint_{C}(2xydx+x^2dy)=\iint_{D}\left(2x-2x\right)dxdy=\iint_{D}0dxdy=0$$

Answer 2

At one point you say, $\oint_C Pdx+ Qdy= -\int_R\int\left(\frac{\partial P}{\partial y}- \frac{\partial Q}{\partial x}\right)dxdy$ is the same as $\oint_C Pdx+ Qdy= \int_R\int\left(\frac{\partial P}{\partial y}+ \frac{\partial Q}{\partial x}\right)dxdy$. I don't know where you got that. At first I thought you were just taking that outside negative into the integral but if you were doing that it would be $\int_R\int\left(-\frac{\partial P}{\partial y}+ \frac{\partial Q}{\partial x}\right) dxdy= \int_R\int\left(\frac{\partial Q}{\partial x}- \frac{\partial P}{\partial y}\right)dxdy$