how to determine the bounds of this integral? 
Let $ G:= \{ (x,y) \in \mathbb{R}^2 \mid x^2 + 4y^2 >1, x^2+y^2 <4 \}$
I want to determine $ \int_G x^2+y^2 d(x,y)$.

So, its the area between a circle with radius 2 und an ellipse with semi-axis 1 in $x$-direction and semi-axis $\frac{1}{2}$ in $y$-direction, right?
How can I determine the bounds? How can I use polar coordinates to transform this?

Let
$$ x = r \cos \theta, \quad y = r \sin \theta. $$
Then since $x^2+y^2 < 4,$ we see that $r < 2$, and since $x^2+4y^2 > 1$, we see that
$$ r^2\cos^2\theta+4r^2\sin^2\theta = r^2+3r^2\sin^2\theta > 1 \quad \iff \quad r > \frac{1}{\sqrt{1+3\sin^2\theta}}. $$
Thus
\begin{align}
\int_G x^2+y^2 \, \mathrm d(x,y) &= \int_0^{2\pi}\int_{\frac{1}{\sqrt{1+3\sin^2\theta}}}^2  r^2 \cdot r \, \mathrm dr \, \mathrm d\theta.\\
\end{align}
Can you proceed?
