How to find double integral with change of variables

Question

I am trying to complete a double integral with change of variables, and am not sure where I am going wrong (somewhere in setting up the integral is my guess). Thank you for any help you can provide!

Use the given transformation to evaluate the integral.

Double integral in the region R of (3xy), where R is the region in the first quadrant bounded by the lines y=1/2x, y=3/2x, xy=1/2, and xy=3/2 and the hyperbolas xy=u/v and y=v.

Answer 1

∫∫_Rxydxdy, where R={y=1/2x, y=3/2x, xy=1/2, and xy=3/2},

First I'm not sure if it is y=(1/2) x or y=1/(2x). The same with y=3/2x,,

I think it should be y=(1/2) x and y=(3/2)x

You can draw, to see how looks the region R. Then make substitution xy=u/v and y=v and find the Jacobian of transformation form (x,y) to (u,v) and define new region. Then evaluate double integral w.r. to u and v.

To find Jacobian, you have to express x and y as functions of u and v, that is x=x(u,v), y=y(u,v):

x=u/v², y=v. Remember how to find Jacobian dxdy=|J| dudv, J={{dx/du,dx/dv},{dy/du,dy/dv}} <-- this a matrix.

Continue, if it is still hard, let me know, I can provide more assistance.