given that vector filed F= < 2x+ sin y, xcosy>, find integral F.dr where C is the arc y=1-x^2 from (0,1) to (1,0).

my working

integral F. dr = double integral curl F dxdy =0.

what is wrong with my workings?

i know that i can make use of F= grad f to solve this question..but im just wondering why my this method doesnt give me the correct answer.