I am having trouble with the following question
Let $\ f:\mathbb{R} \to \mathbb{R}$ be a continuous function, $\ 0 < p < q,$ $\ 0 < r < s,$ and
$\ E=\{(x,y): px\leq {x}^{2}+{y}^{2} \leq qx,\ ry\leq {x}^{2}+{y}^{2} \leq sy\} \subset \mathbb{R}^{2},$
Show that one can define a function $\ F:\mathbb{R}^{2} \to \mathbb{R}$ so that
$$\int_E \ f(y/x) \, dx\ dy\ =\int_{p}^{q} \int_{r}^{s} \ F(u,v)\ du\ dv$$
For all choices of $\ p, q, r,\ $and $\ s.\ $ Find $\ F$ explicitly in terms of $\ f. \ $ Contrive some examples for $\ f \neq \ 0$ to compute this integral without too much work.
Am I suppose to try to make use of polar coordinates to transform the following set of relations $\ p\leq {x} + \frac{{y}^{2}} {x} \leq q,\ r \leq \frac{{x}^{2}} {y} + y \leq s\,$ from $\ (x,y)\ $ to $\ (u,v)=(\frac{y} {x}) \ $ and at the same time eliminate $\theta\ $. From the way the set $\ E\ $ is defined,geometrically, I am not sure if I am integrating over the region of an annulus or the difference in area between two concentric circles.
Thank you in advance.
A point $(x,y)$ belongs to $E$ iff $\>x^2+y^2=v x$, or $\bigl( x-{v\over2}\bigr)^2+y^2={v^2\over 4}$, for some $v\in[p,q]$ and $x^2+y^2=u y$ for some $u\in[r,s]$. Each of these equations describes a circle having its midpoint on a coordinate axis and touching the other axis at the origin. Given $u$ and $v$ the other point of intersection of these two circles computes to $$x={u^2 v\over u^2+v^2},\quad y={u v^2\over u^2+v^2}\ .$$ In other words, the map $$\Psi:\quad R\to E,\qquad(u,v)\mapsto (x,y):=\left({u^2 v\over u^2+v^2}, \>{u v^2\over u^2+v^2}\right)$$ maps the rectangle $R:=[r,s]\times[p,q]$ bijectively onto $E$. Since the Jacobian of $\Psi$ is given by $$J_\Psi(u,v)={u^2 v^2\over(u^2+v^2)^2}$$ it follows that $$\int_E f(y/x)\>{\rm d}(x,y)=\int_R f(v/u)\>J_\Psi(u,v)\>{\rm d}(u,v)=\int_R F(u,v)\>{\rm d}(u,v)$$ with $$F(u,v)=f(v/u){u^2 v^2\over(u^2+v^2)^2}\ .$$