Solve the double integral $\int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)dxdy\:$

Question

$$\int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)\,dx\,dy.$$ I think you need to be solved by the transition to polar coordinates:

\begin{cases} x=r\cos(\phi),\\ y=r\sin(\phi) \end{cases}

Answer 1

Suppose $x=r\cos(\phi)$ and $y=\frac{1}{2}r\sin(\phi)$. Then the Jacobian is $\frac{1}{2}rdrd\phi$. Then $$ \int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)\,dx\,dy= \int_{0}^{2\pi}\int_{0}^{1}(...)\frac{1}{2}rdrd\phi. $$