I need to compute $\iint_K 2xdxdy$ where $K=\left\{(x, y)\in\mathbb{R}^2; \frac{x^2}{4}+y^2\leq1\right\}$.
In my solution, I put $x=2\rho\cos\theta$ and $y=\rho\sin\theta$ and in order to cover $K$, I think that $0\leq \rho\leq 2$ and $0\leq\theta\leq 2\pi$
But taking a look at the solution's manual of the book where this question comes from, the author puts $0\leq \rho\leq 1$. Can anyone explain why?