Integrating exponential function with elliptic bounds

Question

I am trying to integrate the following:

$$\iint_R\exp\left(\frac{x^2}{4}+\frac{y^2}{16}\right)\:\mathrm{d}A$$

With the region $R$ having the bounds:

$$\frac{x^2}{4}+\frac{y^2}{16}=3$$ $$\frac{x^2}{4}+\frac{y^2}{16}=5$$

And I am completely stuck with where to begin on this. I am aware that it involves polar coordinates but even when attempting that it seems to get unbelievably messy. Any help would be greatly appreciated.

Answer 1

First do a change of variables $y = 2 u$. Then express $(x,u)$ in polar coordinates.

Answer 2

Try elliptical coordinates instead, that is: $$x = 2r \cos \theta $$ $$y = 4r \sin \theta $$

Then

$$ \frac{x^2}{4} + \frac{y^2}{16} = r^2 $$

and

$$ \frac{\partial (x, y)}{\partial (r, \theta)} = 8r $$

The integral becomes

$$ \int_0^{2\pi} \int_{\sqrt{3}}^{\sqrt{5}} e^{r^2} 8r \,dr \,d\theta $$