Integrate[(1/(2π (σ^2)))(E^(-((((u^2)+(v^2)))/(2 (σ^2))))),{u,-r,r},{v,-(Sqrt[(r^2)-(u^2)]),(Sqrt[(r^2)-(u^2)])},Assumptions->{r>0}]

Clear[F];

F[x_]=Integrate[(1/(2π (σ^2)))(E^(-((r^2)/(2 (σ^2)))))r,{r,0,(Sqrt[x])},{θ,0,2 π}]

F[0]

Limit[F[x],x->∞, Assumptions ->{σ>0}]