Doubt in Computing $\iint_S x dxdy$ where $S: x^2+y^2=1$

Question

Computing $\iint_S x dxdy$ where $S: x^2+y^2=1$

In text book it is directly written as 0. Any shortcut for this?

I did it in following way.

Given integral = $\iint_{r= 0to1, \theta : 0 to 2 \pi} r \cos \theta r dr d\theta = 0$