$\oint_{\Gamma}^{} (x^{\frac{4}{3}}+y^{\frac{4}{3}})ds,\Gamma \text{ is } x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ my process is below
$$\left\{\begin{matrix} x=r\text{cos}\theta \\ y=r\text{sin}\theta \end{matrix}\right. \ \theta \in [0,2\pi]$$ $\Rightarrow$ $$I=a^\frac{4}{3}\int_0^{2\pi}(\cos^4\theta+\sin^4\theta)3a\sin\theta\cos\theta d\theta$$ $\Rightarrow$ $$3a^\frac{7}{3}\left(-\int_0^{2\pi}\cos^5d(\cos\theta)+\int_0^{2\pi}\sin^5d(\sin\theta)\right)$$ it is zero, but the answer is $4a^\frac{7}{3}$ The answer uses symmetry to limit the range between $0$ and $\frac{\pi}{2}$. Why is my answer wrong and should use symmetry to limit the range first?
And what kind of problem should I limit the range first?