I have to find the volume under $z=x^2+y^2$ and above $4x^2+y^2=1$.
I know $4x^2+y^{2}$ is an ellipse, so I thought about, in polar coordinates $$x=\frac{r\cos(t)}{2}$$ and $$y=r\sin(t)$$ with $0\leq r\leq1,0\leq t\leq 2\pi$. Then $$z=\frac{r^{2}\cos^{2}(t)}{4}+r^{2}\sin^{2}(t)=1-\frac{3r^{2}\cos^{2}(t)}{4}$$ And so, the volume would be $$\int_{0}^{2\pi}\int_{0}^{1}\left(1-\frac{3r^{2}\cos^{2}(t)}{4}\right)rdrdt$$ Is it correct?