The Generalized Stokes Theorem. If $\omega$ is any smooth $(k-1)$ form on $X$, then $$\int_{\partial X} \omega = \int_X d\omega.$$
Let $C \subset \mathbb{R}^2$ be a (smooth) simple closed curve in the plane and $\Omega$ the closed subset that it bounds. Thus $\Omega$ is a $2$-manifold with boundary $C$. Assume the origin, $O \in \mathbb{R}^2$, lies in the interior of $\Omega$.
By Stokes Theorem, $$\iint_{\Omega}r dr \wedge d\theta = \int_C \frac{1}{2}r^2 d\theta.$$
But how can I justify the integrand is well-defined, and what is the orientation?
Thank you!
The curve $c$ is oriented by the direction it is parametrized in. Depending on this orientation, there is a unique orientation on $\Omega$ such that the induced orientation on the boundary $c$ coincides with this parametrization.
Stoke's theorem holds whenever the form $\omega$ is $C^1$. That said, the form $r^2d\theta$ is in fact smooth, as $$ r^2d\theta = x dy - y dx.$$ Its differential $rdr \wedge d\theta$ is in fact the standard volume form on $\mathbb{R}^2$. So no differentiability problems arise in the first place.
\Edit: I will be more elaborate on both points.
1.) A regular curve in the plane is something different than an immersed $1$-fold. A $1$-fold defined by a curve $c$ comes with a natural orientation defined by saying that the basis $\dot{c}(t)$ of the $1$-dimensional tangent space is positively oriented. On the other hand, for the Stokes formula, this is quite irrelevant: You just have to make sure that the orientations of $\Omega$ and the bounding curve fit together in the sense you learned it when talking about the orientation that a manifold induces on its boundary.
2.) If you ask if the integrand is well-defined, shouldn't you know what well-defined means in the first place? The only thing that I can say now is somewhat "philosophical": What exactly is meant by the phrase "We define the one-form $r d \theta$ on $\Omega$", when clearly $d\theta$ is not defined on all of $\Omega$. There could be different interpretations: For example, we defined a $1$-form on almost all on $\Omega$ (everywhere except on a zero set) which suffices for integration. Or, it should be understood that we mean the smooth $1$-form that is defined on all of $\Omega$ and coincides with $r^2d\theta$ everywhere where the latter is defined. This $1$-form exists as I showed you and this seems to me the natural interpretation.