Please help me clarify the following: I want to verify the area of the unit disk $D$ in $\mathbb{R^2}$ by means of the integral $\int_D 1\,dxdy$.
The polar coordinates provide a $C^1$-diffeomorphism from $D-\{x\leq0\}$ to $\{(r,\phi) \mid 0< r< 1,\, 0< \phi < 2\pi\}$, i.e. not the complete domain $D$.
Does this mean that I can't use polar coordinates here since they can't be used on the whole domain $D$? Or is it possible to argue that the integral is unchanged when $\{x\leq0\}$ is removed?