I am trying to solve the following question.
Let C be a simple closed curve in the plane with positive orientation bounding domain D in the plane. Which of the following integrals below is equal to the area of D?
I found that the answer to the above question is $\int_C (2xy + y) dx + (x^2 + 3x) dy = \int_C Pdx + Qdy = \iint_D \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} dA = \iint_D (2x+3) - (2x+2) dA = \iint_D 1 dA$
(Please correct me if I am wrong above.)
But I am not sure how to prove whether $ \int_C x ds $ equates to $\iint_D 1 dA$ or not.
To be exact, I am not sure of how to deal with ds and turn that into an integral with dx or dy to use Green's Theorem.
Thanks for helping out!