I'm trying to determine whether the following double integral converges
$$ \iint_D e^{-x^2-4y^2} dxdy \qquad , \qquad D = [(x,y) : 0 \leq x \leq 2y] $$
So I tried the following substitution
$$ x = r \cdot \text{cos} \theta $$ $$ y = \frac{1}{2} r \cdot \text{sin} \theta$$ $$ \frac{\pi}{6} < \theta < \frac{\pi}{2} $$
But I'm getting a wrong answer $\frac{\pi}{12}$, the right answer is $\frac{\pi}{16}$, but I can't really find where i went wrong! Any tips?