an area $R$ is bounded by the hyperbolas $x^2-y^2=2$, $x^2-y^2=4$, $xy=2$ and $xy=5$ in the first quadrant.
I want to use a substitution to calculate the mass $m=\iint_R \delta(x,y)dA$ of R. It is given that the density $\delta(x,y)$ is proportional to the square of the distance to origo, with proportionality constant $5$.
I've tried to use the transformation $u=x^2-y^2$ and $v=xy$ which gives me the rectangle in the $uv$-plane bounded by: $2\leq u \leq4$ and $2\leq v \leq 4$
I found the jacobi determinant: $\frac{\partial(u,v)}{\partial(x,y)} = 2x^2+2y^2 = 2(x^2+y^2)$. I should be able to express this using my $u$ or $v$ right (I dont see how atm)? Have I done something wrong trying to find the jacobi?
My next step would be to take the inverse to find $\frac{\partial(x,y)}{\partial(u,v)}$. And then use this in the final integral
Hint...Your jacobian is $$\frac{1}{2(x^2+y^2)}$$
Therefore the term $(x^2+y^2)$ doesn't need to be expressed in terms of $u$ amd $v$ because it will be cancelled thanks to the given density expression, and you are left with a very simple integral to do.