Integrating over region bounded by two ellipses

Question

Let $D$ be the region in the first quadrant which is bounded by the ellipses $4x^2 + y^2 = 16$ and $4x^2 + y^2 = 1$. Calculate $$\int\int_D\frac{x}{4x^2 + y^2}dA $$

What would be the best way to approach this problem? Preferably no elliptic coords as those haven't been covered in our class. Im thinking a change of variables could be a good choice. If you decide to do one please explain how you came up with it.

Answer 1

As the second ellipse is all inside the first one, you have $$ \iint_D\frac{x}{4x^2 + y^2}dA= \iint_{E_1}\frac{x}{4x^2 + y^2}dA- \iint_{E_2}\frac{x}{4x^2 + y^2}dA. $$

Answer 2

I suggest the transformation $x=\frac12r\cos\theta$, $y=r\sin\theta$, so that $4x^2+y^2=r^2$. Your integral is then equal to $$\int_0^{\pi/2}\int_1^4{r\cos\theta\over 2r^2}\frac r2\,dr\,d\theta = \frac14\int_0^{\pi/2}\int_1^4\cos\theta\,dr\,d\theta.$$