$limlimits_{Rto0^+}intlimits_{x^2+y^2le R^2}e^{-x^2}cos(y)dxdy=?$

$$limlimits_{Rto0^+}intlimits_{x^2+y^2le R^2}e^{-x^2}cos(y)dxdy=?$$

First I want to show $f(x,y)=e^{-x^2}cos(y)$ doesn’t go crazy at $(0,0)$ otherwise it is already clearly continuous and bounded.

So $$|e^{-x^2}cos(y)|le e^{-x^2}to 1$$ when $$x^2+y^2to 0$$

So Now finite and bounded integrand’s integral has to go 0 because region vanishes, but how to properly show it?