Finding mass of enclosed region in first quadrant, change of variables

Question

The hyperbolas $$ x^2 - y^2 = 1, x^2 - y^2 = 6, xy = 3, xy = 8 $$ enclose a region $R$ in the first quadrant. Use a change of variables to calculate the mass $$ m = \int \int_R \delta(x,y)dA $$ of R, when its given that the density $\delta(x,y)$ is proportional to the square of the distance from the origin, with a proportionality constant of $2$.

I figured $u = x^2 - y^2$ and $v = xy$ is a good substitution, as this gives us a rectangular region to integrate over. I suppose the integrand should be $2\sqrt{x^2 + y^2}$. Not sure how to get this in terms of $u$ and $v$ though. Also not sure how to compute the jacobian in this case.

Answer 1

HINT

• the substitution is a very good idea
• the integrand is $2(x^2 + y^2)$ (twice the square of the distance)
• the Jacobian is

$$|J|=\begin{vmatrix}u_x&u_y\\v_x&v_y\end{vmatrix}=\begin{vmatrix}2x&-2y\\y&x\end{vmatrix}=2(x^2+y^2)$$

then

$$m = \int \int_R \delta(x,y)dA=\int \int_R 2(x^2+y^2)dxdy=\int_1^6 \int_3^8dvdu$$