Let $\omega = \frac{x}{x^2+y^2}dy - \frac{y}{x^2+y^2}dx$ is the smooth one-form defined on $\Bbb{R}^2\setminus \{0\}$, has nonzero integral over the $S^1$.
Using stokes's theorem we can prove there is no orientable compact smooth submanifold of $\Bbb{R}^2\setminus\{0\}$ that has boundary $S^1$
I was confused why the closed disk removed one point is not the desired submanifold.
that is consider the $D^2-\{0\}$ which is closed and bounded in $\Bbb{R}^2\setminus \{0\}$.which also has manifold structure.Which has boundary $S^1$?Why this closed disk does not satisfy the condition?
The Heine-Borel property that "compact" is equivalent to "closed and bounded" is true in $\mathbb{R}^n$, but it's not true in any topological or metric space. It's not true in the plane puntcured at zero, and the closed unit disc punctured at zero is most certainly not compact.
PS just to add - a metric space is compact if and only if it's a complete metric space and totally bounded (= has a finite cover by $\epsilon$-balls for any $\epsilon >0$). The punctured disc is not complete (missing origin), although I think it is totally bounded.