Finding the change of variables when they are not specified

Question

I am asked to solve this integral over the area D that lies in the first octant $(x ≥ 0) (y≥0)$. Where D is the area between the two ellipsoids $4x^2 + y^2 = 16$ and $4x^2 + y^2 = 1$. I think using change of variables will help me solve this integral, however I do not know how to do so. I have tried using the trigonometric property $\sin^2(θ) + \cos^2(θ) = 1$, where u is r and v is θ. $$\iint \frac {x}{4x^2+y^2}dA$$

Answer 1

Try the substitution, $u=x$ and $v= 4x^2 + y^2.$

The region of integration transforms to a rectangle, $ u\in [1/2,2],$ and $v\in [1,16].$

Do not forget your Jacobean for $dA$

Answer 2

Switching to polar coordinates is a reasonable idea, but you would do better by choosing a substitution that follows the boundaries of the region. The two ellipses are homothetic and concentric, so something like $x = r\cos\theta$, $y=2r\cos\theta$ for $\frac12\le r\le2$, $0\le\theta\le\frac\pi2$ with Jacobian determinant $2r$ looks promising. This gives you a parameterization of the inner and outer ellipses for $r=\frac12$ and $r=2$, respectively, and conveniently the denominator of the original integrand transforms into a multiple of $r$. The integration is then over a rectangle in $(r,\theta)$-space.