Find a regular, non-constant, closed curve $\gamma:[0,1] \rightarrow \Bbb R^2$ such that $\int_\gamma x^3dy-y^3dx=0$.
In case where $\gamma$ is $C^1$ smooth and is the boundary of a closed set, Green Theorem implies that this is not possible. Any ideas?
Take $\gamma$ af following:$$x=\sin 4\pi t\\y=\sin 2\pi t$$therefore$$\int_{\gamma}x^3dy-y^3dx=2\pi\int_{0}^{1}(\sin^3{4\pi t}\cos 2\pi t-2\sin^3 2\pi t\cos4\pi t)dt\\=2\pi\int_{0}^{1}\dfrac{3\sin{4\pi t}-\sin12\pi t}{4}\cos 2\pi t-\dfrac{3\sin{2\pi t}-\sin6\pi t}{2}\cos4\pi t)dt=0$$since $$\int_{0}^{T}\sin\dfrac{2m\pi}{T}x\cos\dfrac{2n\pi}{T}xdx=0\qquad,\qquad \forall m,n\in\Bbb Z$$A sketch of $\gamma$ is as following: